Although Titus is not recorded specifically by name in the book of Acts, he is mentioned frequently in Paul's epistles. The epistle to Titus is of course written directly to him. See By The Book - Titus.
Titus was with Paul and Barnabas at Antioch, and accompanied them to the council at Jerusalem where Paul defended his ministry to the Gentiles:
"Then after fourteen years I went up again to Jerusalem with Barnabas, taking Titus along with me. I went up by revelation; and I laid before them, but privately before those who were of repute, the gospel which I preach among the Gentiles, lest somehow I should be running or had run in vain. But even Titus, who was with me, was not compelled to be circumcised, though he was a Greek." (Galatians 2:1-3 RSV) (see also Acts 15:1-35).
Some time later, Titus was with Paul and Timothy at Ephesus, from where he was sent by Paul to Corinth to gather contributions from the church there to help the poor saints at Jerusalem:
"For they gave according to their means, as I can testify, and beyond their means, of their own free will, begging us earnestly for the favor of taking part in the relief of the saints - and this, not as we expected, but first they gave themselves to The Lord and to us by the will of God. Accordingly we have urged Titus that as he had already made a beginning, he should also complete among you this gracious work." (2 Corinthians 8:3-6 RSV).
Titus later rejoined Paul in Macedonia:
"And besides our own comfort we rejoiced still more at the joy of Titus, because his mind has been set at rest by you all. For if I have expressed to him some pride in you, I was not put to shame; but just as everything we said to you was true, so our boasting before Titus has proved true. And his heart goes out all the more to you, as he remembers the obedience of you all, and the fear and trembling with which you received him." (2 Corinthians 7:13-15 RSV).
Titus is not again mentioned until a few years later, after Paul's first imprisonment, when he was working with the church in Crete (seen in the photograph):
"To Titus, my true child in a common faith: Grace and peace from God the Father and Christ Jesus our Savior. This is why I left you in Crete, that you might amend what was defective, and appoint elders in every town as I directed you." (Titus 1:4-5 RSV)
The last mention of Titus was when he had left for Dalmatia, after being with Paul at Rome during Paul's second imprisonment. By then, Paul, nearing his martyrdom, was left with only Luke:
"For I am already on the point of being sacrificed; the time of my departure has come. I have fought the good fight, I have finished the race, I have kept the faith. Henceforth there is laid up for me the crown of righteousness, which the Lord, the righteous judge, will award to me on that Day, and not only to me but also to all who have loved his appearing. Do your best to come to me soon. For Demas, in love with this present world, has deserted me and gone to Thessalonica; Crescens has gone to Galatia, Titus to Dalmatia. Luke alone is with me." (2 Timothy 4:6-11 RSV)
Fact Finder: Did Paul tell Titus to always teach what is in accordance with sound doctrine?