The book of Acts gives several examples that repentance and faith are two separate acts and repentance is a prerequisite for salvation. Even though the actual act of repentance is usually not mentioned, it is still evident.
Probably the most clear passage showing that repentance and faith are two separate acts is Acts 20:20-21,
"And how I kept back nothing that was profitable unto you, but have shewed you...Testifying both to the Jews, and also to the Greeks, repentance toward God, and faith toward our Lord Jesus Christ."
This verse is a bane to those who treat the terms as synonymous. It is clear and reasonable to understand "repentance" and "faith" as separate concepts just like "God" and "Jesus Christ" are separate persons of the godhead. Repentance is toward God (the Father) because He is the one harmed by our sin; faith is toward Christ because He is the redemption: the source of salvation. Paul plainly preached both (Act 17:30) and said both were "profitable."
Although repentance is not always directly mentioned, the concept can be found in all the accounts of people getting saved in the book of Acts.
Consider the Ethiopian Eunuch. In Acts 8:34 he asks, "I pray thee, of whom speaketh the prophet this? of himself, or of some other man?" The Eunuch was contemplating a passage in Isaiah 53 dealing with death and was seeking truth. What brought him to that point? At some time in his past, prodded by a guilty conscience most likely (Rom 2:15), he had a consciousness of his guilt before God which turned his mind about his standing with Him. Or possibly, when Phillip spoke of how Jesus was the sheep in the passage who died for our sins, that is when he contemplated his guilt. In either case that was his repentance. He only lacked the knowledge of Christ to be able to believe, and when it came, he quickly believed and was saved (Acts 8:37).
Saul of Tarsus
Saul (later Paul) met the Lord on the Damascus Road while he was traveling to persecute Christians (Acts 9:1-9). When a bright light from heaven shined around him he didn't know what was happening and fell to the earth. When he heard the voice speaking to him he didn't know who it was and asked, "Who art thou, Lord" (Acts 9:5). When the voice identified Himself as Jesus, Paul had a life-changing epiphany; a massive change in his "worldview" and perspective of truth and reality. He instantly realized everything he had been believing and doing was terribly wrong and had a complete change of attitude and direction. This is all evidenced by his trembling words "Lord, what wilt thou have me to do?"
In the seconds that transpired between Acts 9 verses 5 and 6 Saul repented and believed. He changed his outlook on Jesus Christ, his standing before God, and his life. He was astonished at how wrong he had been and that the very Jesus Christ he had been railing against was now speaking to and confronting him. Trembling from the shock of the situation and in sore regret of his past (Acts 22:4, 19-20, 26:9-11), he asked how he could now serve this very same Jesus he persecuted. Something he would have thought laughable just minutes before. The conversion of Saul is an obvious example of repentance.
The account of Cornelius is another obvious example. He was an unsaved man who was told by an angel his alms and prayers were a "memorial before God" (Acts 10:4). He had been very actively seeking God. Was he born this way? Of course not. Through the reproving of the Holy Spirit (John 14) and accusing guilt of his conscience (Rom 2:15), sometime in his past he repented and was determined to seek God and get right with Him the best way he knew how. Like the eunuch, all he lacked was access to the truth of Christ so he could believe more perfectly. When Peter brought the gospel, as soon as they all heard enough to believe, they did believe. Their hearts had already been prepared by their earlier repentance.
Lydia is another example. It is said "she worshipped God" (Act 16:14), probably much like Cornelius, and she had a "heart the Lord opened, that she attended unto the things which were spoken of Paul." Sometime in the past she faced her sins, changed her mind in repentance, and started to worship the Lord as best she could. Like the others, she couldn't believe the gospel until a "preacher" came and told it to her.
The Philippian jailer is a little different example (Acts 16:31). There is no indication he even thought much about God before the night Paul and Silas became "guests" in his jail. The repentance in his case came in the same night he and his family believed, fueled by Paul and Silas not escaping when they could and enhanced by their attitude while in his jail. When the earthquake hit and the prisoners were still there, plus Paul's charge for him not to kill himself, showing he cared for him, the man had an epiphany and drastic turning of mind and sought to be saved in the same moment...and he was. Only a repentant soul will sincerely cry out in trembling desperation (Acts 16:29-30), "What must I do to be saved?" The obvious reason Paul did not mention any need to repent is it was very obvious the man already had!
John the Baptist's Disciples
Then there is John the Baptist's disciples in Acts 19:1-7. They were baptized unto repentance by John and had apparently been living in a repentant state ever since, but they also lacked the truth of Christ. When they received it (most certainly, Paul told them about Christ); they must have believed (even though it is not specifically mentioned) because Paul baptized them again in believer's baptism.
These six accounts show repentance is separate from belief and can even be separated by years from belief. They all line up perfectly with Acts 20:21 and the rest of Scripture and show repentance is essentially a remorseful turning of mind towards God that prepares the heart to truly believe on Christ.