Acts of Apostles in the Early Church 

Luke is a good historian and gives us a large amount of speech material in Acts. It's more than you find in other ancient historians such as Herodotus, Tactius, Polybius, or Thucydides.

Luke seems to have been following the Greek tradition of writing history, a tradition that didn't have a convention of creating fictitious speeches for real historial people in place of actually having information about what they said.

According to Thucydides, it was difficult to retain anything like a verbatim account, but a good historian offered summaries with the major points of what was said (History of the Peloponesian War 1.22.1-2).

In Acts we are dealing with summaries of speeches, except perhaps in the case of Stephen's speech. 

Luke traveled with Paul and could have gotten the material from talking with the apostle, and in a few instances from being present to hear the speeches. But Luke doesn't claim to have directly heard most of the speeches, and so we must assume that he relied on sources--Paul for Stephen's speech, and members of the Jerusalem church for the others.

He likely gathered this material near the end of the 50s A.D. while he was in the area during Paul's two-year incarceration in Caesarea Maritima. It's also possible that Luke wrote the speeches and edited them to reflect his own style of writing at various points.

Speeches

Peter (Acts 1; 2; 3; 4; 5; 10; 11; 15)

James (Acts 15; 21)

Stephen (Acts 7)

Paul (Acts 13; 14; 17; 20; 22; 23; 24; 26; 28)

Non-Christian (Acts 5:35-39; 19:35-40; 24:2-8; 25:14-21, 24-27)

Speeches in Acts by F.F. Bruce