To sanctify yourself it is necessary for you to employ the means, — such as, to avoid evil occasions, to remain detached from earthly goods, to live a life recollected in God; and to maintain this, it is necessary to receive the sacraments frequently, to make your meditation, your spiritual reading, and to perform other devout exercises, everyday, otherwise it is impossible to preserve the spirit of fervour. St Alphonsus Ligouri
What does it mean for a Christian to "sanctify oneself"?
That’s a good question. To “sanctify one’s self” is just another way of saying “growing in holiness.” To grow in holiness, one can’t just live a life that only consists of random opportunities to live a godly life pleasing to the Lord, though that is something we should do. God expects us to strive for holiness:
You, therefore, must be perfect, as your heavenly Father is perfect. Mt 5:48
I am speaking in human terms, because of your natural limitations. For just as you once yielded your members to impurity and to greater and greater iniquity, so now yield your members to righteousness for sanctification. When you were slaves of sin, you were free in regard to righteousness. But then what return did you get from the things of which you are now ashamed? The end of those things is death. But now that you have been set free from sin and have become slaves of God, the return you get is sanctification and its end, eternal life. Romans 6:19:22
Finally, brethren, we beseech and exhort you in the Lord Jesus, that as you learned from us how you ought to live and to please God, just as you are doing, you do so more and more. For you know what instructions we gave you through the Lord Jesus. For this is the will of God, your sanctification…For God has not called us for uncleanness, but in holiness. Therefore whoever disregards this, disregards not man but God, who gives his Holy Spirit to you. 1 Thes 4:1-8
Since we have these promises, beloved, let us cleanse ourselves from every defilement of body and spirit, and make holiness perfect in the fear of God 2 Cor 7:1
For God has not called us for uncleanness, but in holiness. 1Th 4:7
Yet woman will be saved through bearing children, if she continues in faith and love and holiness, with modesty. 1Ti 2:15
Strive for peace with all men, and for the holiness without which no one will see the Lord. Heb 12:14
Since all these things are thus to be dissolved, what sort of persons ought you to be in lives of holiness and godliness. 2Pe 3:11
To positively grow in holiness, one must undertake a plan for holiness. This has nothing to do with piling up works and prayers, or going through the motions. It does, however, involve taking on the exercise of the moral virtues, those attributes that help a mature Christian exercise grow in holiness. Virtues are nothing more than good Christian habits—classically there are seven: Faith, Hope, Charity, Fortitude, Justice, Prudence and Temperance. Just as a Christian can grow in bad habits—called vices—he can also grow in these good spiritual habits, making it more difficult for him to sin and easier for him to do the right.
To grow in holiness, one also needs to develop a detachment from the world. This means not being attached to the fads, opinions, noise and hyperactivity of the prevailing culture. We should question seriously anything in our lives that leads us away from God. For example: do I neglect to pray because I can’t miss my favorite show, however innocent and even spiritually edifying it might be? Then my favorite show is drawing me away from God. If I can watch my show and pray, then that’s OK. Laypersons like ourselves can never grow completely detached from the world because we live in it everyday, but to the extent that we can, we should.
Speaking of prayer, we won’t grow in holiness without regular prayer. I know you know this, so I won’t belabor it, but we both know it isn’t an option for Christians.
We must leave ourselves open to and take advantage of opportunities for grace. Just being a Christian means, assuming we are in a state of sanctifying grace (i.e. not having any unrepented/unconfessed mortal sins) we have the Holy Spirit in us always, but there are special moments of actual grace that God showers on us. As a Catholic, of course, for me the Sacraments (especially the Eucharist and Confession) are means of grace since they are an encounter with Christ. For married persons, there are also the graces given in the Sacrament of Matrimony.
Finally, one shouldn’t neglect spiritual reading like devotionals, lives of the saints, meditations, and especially the Scriptures. Just like our bodies are trained by the virtues and our hearts are softened by prayer, our minds must be fed with the wisdom of the saints, and the pure word of God.
Hope that helps. I found a good article by Mother Angelica of EWTN who talks about this very subject. Maybe this can be your spiritual reading for today!
You might also find the article "Grace: What It is and What it Does," from Catholic Answers, helpful as well.