In the early church, it was unheard of for a believer to not be baptized (see Acts 8:36, 38). Like other believers at that time, Paul was baptized shortly after being saved. While Acts 22:16 sounds as if he was saved because he was baptized, there are a couple key arguments against that understanding. First, that same verse connects the washing of his sins to “calling on Jesus’ name” (Acts 22:16b). That is, the expression of his faith was the key part to his salvation and not the baptism. Second, in the historical account in Acts 9, we see him acting like, and being called, a believer before the Baptism. See, in particular, Acts 9:11 as well as how Ananias refers to him as a brother in Acts 9:17. Finally, the rest of the New Testament, a large portion of which was written by Paul, teaches that salvation is by faith alone (Ephesians 2:8–9).
Baptism does not save. Instead, it is an expression of one’s saving faith. However, though it does not save, it is also expected to be done by all believers. There may be times when wisdom leads to delaying baptism after one’s confession. However, that should be the exception.
When Jesus gave His final commands to the disciples, He told them to do two things after disciples were made: they were to be baptized and taught (Matthew 28:19–20). His intention was that all disciples (everyone who is saved) would also be baptized. He commanded that because it is an act which publicly shows a believer’s allegiance belongs to the Lord (Romans 6:3–4).
So, though baptism is closely associated with salvation, it comes after it. Salvation, itself, comes only by faith and not works, not even the “work” of baptism (Ephesians 2:8–9). This means that of first importance is repentance for sin. Everyone is born an enemy of God and under His wrath (Romans 3:9–18, John 3:36). Short of faith in Christ, everyone is waiting for that final judgement to be unleashed (Romans 5:9). Therefore, it is imperative that everyone repents now. That expression of faith is what saves.
However, once a man or woman does repent, then the very next step that Jesus expects of them is the declaration of their new faith to the world by being baptized. Baptism is an act of joy as much as it is obedience to Christ. He, and He alone, brings salvation. Being baptized is the first action we can do to tell others about Jesus, just like it was the first one that Paul did, shocking both believers and unbelievers because of his complete reversal from persecutor to champion of Christ (Acts 9:19b–22).