It seems from the stories in Acts that after Paul had been arrested, the Romans thought the charges were without foundation. Why didn't they just release him?
The obvious answer is "politics, politics, and more politics!" Yet, it is a bit more complicated than that. Let's look at each Roman in turn. The first would have been the arresting officer, Claudius Lysias, the Roman commander in charge of the garrison attached to the temple. When he heard the noise coming from the temple, he immediately thought of a revolution. Jewish nationalism always intensified during feast days, and he would have already been on high alert. Plus, at that time, the Romans were on the lookout for a potential revolutionary from Egypt, who was gathering a large following. Lysias might have thought Paul was that guy. Arresting him would have been quite a coup for any Roman. When Paul said he was a citizen of Tarsus, the commander's hopes of a promotion vanished. He then instructed his men to torture him until he told the truth (a common practice). When Paul said he was a Roman citizen, flogging was no longer an option. The commander was then placed in the awkward position of having to protect Paul. The Sanhedrin was no help; if anything it was the problem. When Lysias heard about the assassination plot, he had to get Paul out of there. Yet the letter he wrote to the governor clearly stated that Paul had done nothing worthy of death or imprisonment.
Governor Felix was the recipient of that letter. He would hold Paul in custody for two years. According to Acts 24:27, he did this to placate the Jews. Maybe. By reputation, however, Felix was not the placating type. If anything, he might have kept Paul in custody simply because the Jews wanted him released. We can't know that for sure, but there are many instances in which he went out of his way to make life difficult for everyone in his province. Some scholars think he kept Paul imprisoned because he hoped Paul would offer him a bribe. That would be more consistent with his style of governing. Another possibility, however, lies with his wife; Drusilla was a Jewess. It states that Paul talked to them about "faith in Christ." Their marriage had occurred under dubious circumstances. It is not likely that Paul minced words about their shortcomings in terms of salvation. Scholars have argued that Felix kept Paul confined (and alive) because Drusilla was afraid of harming him in any way, lest they bring the wrath of God down upon themselves. Whatever the reason, Felix sat on the case for two years.
Felix was replaced by Festus, who actually did try to make peace with the various ethnic groups in his province. Shortly after he had been appointed, he went around listening to people's grievances and problems. The Jewish leaders wasted no time in letting him know they wanted Paul back for trial. But, after listening to Paul, he also knew that it did not involve matters of state, but of religion. When he asked if Paul would be willing to stand trial in Jerusalem on matters of religion, he naively thought Paul could get a fair trial there. He didn't have the whole picture, but then, he had only been in his position for a few weeks. He didn't yet know the deep-seated hatred for Paul among the members of the Sanhedrin. When Paul appealed to Caesar, he responded in a way that suggests he might have thought it to be overkill. Yet, the request had been made, and it had to be honored.
When Agrippa II and his sister showed up, they all listened to Paul. Festus thought Paul was delusional (and harmless), but Agrippa was impressed. Together, they both agreed that Paul had done nothing worthy of death or imprisonment. Yet, they were duty bound to send him to Rome for adjudication of his case. (To our knowledge, however, no such meeting ever happened before Caesar.) Paul's request to see Caesar might have been in response to his vision that said he would end up in Rome. We can't know that for sure, either. Nonetheless, if Paul had not made the request, he might have been a free man. On the other hand, if he had agreed to go back to Jerusalem for trial, he might not have survived the trip.