Since you are a SAINT, you have a responsibility to live a saintly, sanctified, holy life, making God glorifying choices (not keeping lists of do's & don'ts)  so that you "walk in a manner worthy of the calling with which you have been called," (see note Ephesians 4:1)

Pursue literally means to run swiftly in order to catch a person or thing, run after, follow after, press on: Figuratively dioko describes one who in a race runs swiftly to reach the goal. The idea is to strive, to seek after eagerly, and to earnestly endeavor to acquire. Implicit is the idea that this involves intense personal effort.  The verb tense and mood is present imperative which commands one to pursue peace and holiness as their lifestyle. This verse can be misinterpreted to imply that we are to "work for" our salvation and that if we don't we will lose it. That is not what the writer is saying, nor does the New Testament support such a teaching. See the following note for an enlightening explanation of this potentially confusing verse.

William MacDonald provides one of the most lucid, Scripturally reasoned explanations of Hebrews 12:14, a verse that can easily be misinterpreted, leading to confusion and discouragement regarding our "great salvation".

MacDonald writes...

We should also strive for the holiness without which no one will see the Lord. What is the holiness referred to here? To answer the question we should remind ourselves that holiness is used of believers in at least three different ways in the NT.

First of all, the believer becomes positionally holy at the time of his conversion; he is set apart to God from the world (1Co 1:2; 6:11). By virtue of his union with Christ, he is sanctified forever. This is what Martin Luther meant when he said, “My holiness is in heaven.” Christ is our holiness, that is, as far as our standing before God is concerned.

Then there is a practical sanctification (1Th 4:3; 5:23). This is what we should be day by day. We should separate ourselves from every form of evil. This holiness should be progressive, that is, we should be growing more and more like the Lord Jesus all the time.

Finally, there is complete or perfect sanctification. This takes place when a believer goes to heaven. Then he is forever free from sin. His old nature is removed, and his state perfectly corresponds to his standing.

Now which holiness are we to pursue? Obviously it is practical sanctification that is in view. We do not strive after positional sanctification; it is ours automatically when we are born again. And we do not strive after the perfect sanctification that will be ours when we see His face. But practical or progressive sanctification is something that involves our obedience and cooperation; we must cultivate this holiness continually. The fact that we must follow it is proof that we do not fully attain it in this life.

Wuest writes:

The exhortation is to the born-again Jews who had left the Temple, to live such consistent saintly lives, and to cling so tenaciously to their new-found faith, that the unsaved Jews who had also left the Temple and had outwardly embraced the New Testament truth, would be encouraged to go on to faith in Messiah as High Priest, instead of returning to the abrogated sacrifices of the Levitical system. These truly born-again Jews are warned that a "limping Christian life" would cause these unsaved Jews to be turned out of "the way".

But a difficulty remains! Is it true that we cannot see the Lord without practical sanctification? Yes, there is a sense in which this is true; but let us understand that this does not mean that we earn the right to see God by living holy lives. Jesus Christ is our only title to heaven. What this verse means is that there must be practical holiness as a proof of new life within. If a person is not growing more holy, he is not saved. When the Holy Spirit indwells a person, He manifests His presence by a separated life. It is a matter of cause and effect; if Christ has been received, the rivers of living water will flow. (MacDonald, W & Farstad, A. Believer's Bible Commentary: Thomas Nelson or Logos) (Bolding added)

C H Spurgeon commenting on Hebrews 12:14 wrote that...

Jesus prayed to His Father beseeching Him to make His disciples holy...

IN THE TRUTH:

Israel forgot that she was God's "people, a treasured possession," (KJV "a peculiar people" Dt 26:18, 19)  a "holy nation" (Ex 19:6) and began to break down the walls of separation that made her special and distinct. God commanded them to put a “distinction between the holy and the profane, and between the unclean and the clean” (Lev 10:10); but they ignored the differences and disobeyed God. They became like all the corrupt idolatrous pagan nations around them and it ultimately destroyed them. (devotional) So too the church is of most value to God when it is least like the world in which it exists to be an ambassador of reconciliation. Come out from the world & be separate is the ageless, clarion call finally fully obeyed by the bride who makes herself ready for her Bridegroom with the righteous acts of the saints (Re 19:7-note; Re 19:8-note).

Ray Stedman (from the Message of First Peter) associates "holiness" with "wholeness" adding that

A FEW QUOTES ON
SANCTIFICATION & HOLINESS

A. W. Tozer wrote

Were some watcher or holy one from the bright world above to come among us for a time with the power to diagnose the spiritual ills of church people, there is one entry which I am quite sure would appear on the vast majority of his reports: Definite evidence of chronic spiritual lassitude; level of moral enthusiasm extremely low.

Dwight L. Moody (See Why God Used Dwight L. Moody by R. A. Torrey) once said that...

A holy life will make the deepest impression. Lighthouses blow no horns, they just shine.

Hudson Taylor, famed missionary to China wrote:

"We are not only to renounce evil, but to manifest the truth. We tell people the world is vain; let our lives manifest that it is so. We tell them that our home is above and that all these things are transitory. Does our dwelling look like it? O to live consistent lives!"

John Brown, 19th-century Scottish theologian:

"Holiness does not consist in mystic speculations, enthusiastic fervors, or uncommanded austerities; it consists in thinking as God thinks, and willing as God wills.”

C. H. Spurgeon had much to say about holiness...

Though you have struggled in vain against your evil habits, though you have wrestled with them sternly, and resolved, and re-resolved, only to be defeated by your giant sins and your terrible passions, there is One who can conquer all your sins for you. There is One who is stronger than Hercules, who can strangle the hydra of your lust, kill the lion of your passions, and cleanse the Augean stable of your evil nature by turning the great rivers of blood and water of his atoning sacrifice right through your soul. He can make and keep you pure within. Oh, look to him!

Holy desires are grace in the blade, and the heavenly Husbandman will cultivate them till they come to the full corn in the ear. God-fearing men desire to be holy, to be useful, to be a blessing to others, and so to honor their Lord. - Faith's Checkbook

If we divide ourselves between God and mammon, or Christ and self, we shall make no progress. We must give ourselves wholly to holy things, or else we shall be poor traders in heavenly business; and at our stock-taking, no profit will be shown.  Faith's Checkbook

You will not gain holiness by standing still. Nobody ever grew holy without consenting, desiring, and agonizing to be holy. Sin will grow without sowing, but holiness needs cultivation. Fol­low it; it will not run after you. You must pursue it with determination, with eagerness, with perseverance, as a hunter pursues his prey.
 

The bloom of the hawthorn or White May looks like snow out in the country, but near the vast city or along the roadside its virgin whiteness is sadly stained. Too often contact with the world has just such an effect on our piety. We must make our way to the far-off garden of Paradise to see holiness in its unsullied purity, and meanwhile we must be much alone with God if we would maintain a gracious life below

There can be no such thing as perfect happiness till there is perfect holiness.

I would sooner be holy than happy if the two things could be divorced. Were it possible for a man always to sorrow and yet to be pure, I would choose the sorrow if I might win the purity, for to be free from the power of sin, to be made to love holiness, is true happiness.

A faith which works not for purification will work for putre­faction. Unless our faith makes us pine after holiness, it is no better than the faith of devils, and perhaps it is not even so good as that. A holy man is the workmanship of the Holy Spirit.

We say of a river that it runs to the south, although there may be eddies along the banks which run in an opposite direction to the main stream. Still, these are inconsiderable matters. The main stream of the Thames is running constantly toward the sea, and we speak not untruthfully when we say that it is so. And the main stream and set of the current of the life of a child of God runs toward that which is right and true and holy, both toward God and man. If it is not so with you, you do not know the Lord. You need to be born again...

Of all the griefs the church ever feels, the keenest is when those who once stood in her midst dishonor the name of Christ by unholy living.

Did you ever see a bush burn, and yet not be consumed? Did you ever see a spark float in the sea, and yet not be quenched? Many persons here are, to themselves, just such wonders. They are living godly lives in the midst of temptation, holy in the midst of impurity, serving God in spite of all opposition. They are strange things!.

If your religion does not make you holy, it will damn you. It is simply painted pageantry to go to hell in.

Holiness is better than moral­ity. It goes beyond it. Holiness affects the heart. Holiness re­spects the motive. Holiness re­gards the whole nature of man. A moral man does not do wrong in act; a holy man hates the thought of doing wrong. A moral man does not swear, but a holy man adores. A moral man would not commit outward sin; a holy man would not commit inward sin. And if committed, he would pour forth floods of tears.

"While the Austrian general was staying at the Hotel de Ville, upon the Grand Canal at Venice, I lodged at the same house, and as often as I passed his rooms, whether during the day or at night, I encountered two sentries on guard at the door. My heart said to itself, whenever the King of kings deigns to make a chamber of my spirit, let me set holiness and devotion to be sentries at the entrance. When our Beloved visits us He must not be disturbed. Ill thoughts must be repulsed, and carnal desires kept at a distance. With drawn swords let watchfulness preserve the sanctity of Immanuel's rest. "I charge you, O ye daughters of Jerusalem, by the roes and by the hinds of the field that ye stir not up nor awake my love, till he please." "For the love of Christ controls us, having concluded this, that one died for all, therefore all died;" (2 Cor 5:14)

“In that day shall there be upon the bells of the horses, HOLINESS UNTO THE LORD”—Zechariah 14:20 HAPPY day when all things shall be consecrated, and the horses’ bells shall ring out holiness to the Lord! That day has come to me. Do I not make all things holy to God? These garments, when I put them on or take them off, shall they not remind me of the righteousness of Christ Jesus, my Lord? Shall not my work be done as unto the Lord? Oh, that today my clothes may be vestments, my meals sacraments, my house a temple, my table an altar, my speech incense, and myself a priest! Lord, fulfill thy promise, and let nothing be to me common or unclean.

Let me in faith expect this. Believing it to be so, I shall be helped to make it so. As I myself am the property of Jesus, my Lord may take an inventory of all I have for it is altogether His own; and I resolve to prove it to be so by the use to which I put it this day. From morning till evening, I would order all things by a happy and holy rule. My bells shall ring: why should they not? Even my horses shall have bells: who has such a right to music as the saints have? But all my bells, my music, my mirth shall be turned to holiness and shall ring out the name of “The Happy God.” - Faith's Checkbook

Spurgeon commenting on the truth that God "saved us and called us with a holy calling" (2 Timothy 1:9) wrote that...

The apostle uses the perfect tense (completed action at a point in time in the past with continuing effect/result) and says, "Who hath saved us." Believers in Christ Jesus are saved. They are not looked upon as persons who are in a hopeful state, and may ultimately be saved, but they are already saved. Salvation is not a blessing to be enjoyed upon the dying bed, and to be sung of in a future state above, but a matter to be obtained, received, promised, and enjoyed now. The Christian is perfectly saved in God's purpose; God has ordained him unto salvation, and that purpose is complete. He is saved also as to the price which has been paid for him: "It is finished" was the cry of the Saviour ere he died. The believer is also perfectly saved in his covenant head, for as he fell in Adam, so he lives in Christ. This complete salvation is accompanied by a holy calling. Those whom the Saviour saved upon the cross are in due time effectually called by the power of God the Holy Spirit unto holiness: they leave their sins; they endeavor to be like Christ; they choose holiness, not out of any compulsion, but from the stress of a new nature, which leads them to rejoice in holiness just as naturally as aforetime they delighted in sin. God neither chose them nor called them because they were holy, but he called them that they might be holy, and holiness is the beauty produced by his workmanship in them. The excellencies which we see in a believer are as much the work of God as the atonement itself. Thus is brought out very sweetly the fulness of the grace of God. Salvation must be of grace, because the Lord is the author of it: and what motive but grace could move him to save the guilty? Salvation must be of grace, because the Lord works in such a manner that our righteousness is for ever excluded. Such is the believer's privilege-a present salvation; such is the evidence that he is called to it-a holy life. (Morning and Evening)

Spurgeon commenting on Jesus' command to "sanctify them through Thy truth" (John 17:17) reasoned that...

Sanctification begins in regeneration. The Spirit of God infuses into man that new living principle by which he becomes "a new creature" in Christ Jesus (Ed note: He is "positionally" holy in Christ at this point in time). This work, which begins in the new birth, is carried on in two ways-mortification, whereby the lusts of the flesh are subdued and kept under; and vivification, by which the life which God has put within us is made to be a well of water springing up unto everlasting life. This is carried on every day in what is called "perseverance," by which the Christian is preserved and continued in a gracious state, and is made to abound in good works unto the praise and glory of God; and it culminates or comes to perfection, in "glory," when the soul, being thoroughly purged, is caught up to dwell with holy beings at the right hand of the Majesty on high. But while the Spirit of God is thus the author of sanctification, yet there is a visible agency employed which must not be forgotten. "Sanctify them," said Jesus, "through thy truth: thy word is truth." The passages of Scripture which prove that the instrument of our sanctification is the Word of God are very many. The Spirit of God brings to our minds the precepts and doctrines of truth, and applies them with power. These are heard in the ear, and being received in the heart, they work in us to will and to do of God's good pleasure. The truth is the sanctifier, and if we do not hear or read the truth, we shall not grow in sanctification. We only progress in sound living as we progress in sound understanding. "Thy word is a lamp unto my feet and a light unto my path." Do not say of any error, "It is a mere matter of opinion." No man indulges an error of judgment, without sooner or later tolerating an error in practice. Hold fast the truth, for by so holding the truth shall you be sanctified by the Spirit of God. (Morning and Evening)