There are a couple passages in Acts that speak strongly against the idea that the disciples believed in only a spiritual resurrection of Jesus, and not a physical one. (These are by no means the only reasons to think Jesus’ resurrection was physical, but I rarely hear these passages pointed out in connection with the subject, and they’re worth noting.)
The first is the words of Peter’s evangelistic sermon in Acts 2:22–36:
[Y]ou nailed [Jesus] to the cross by the hands of godless men and put Him to death. But God raised Him up again, putting an end to the agony of death, since it was impossible for Him to be held in its power. For David says of Him, “...You will not abandon my soul to Hades, nor allow Your Holy One to undergo decay.”
In case they missed the fact that Jesus’ body did not decay, Peter continues:
Brethren, I may confidently say to you regarding the patriarch David that he both died and was buried, and his tomb is with us to this day. And so, because he was a prophet...he looked ahead and spoke of the resurrection of the Christ, that He was neither abandoned to Hades, nor did His flesh suffer decay. This Jesus God raised up again, to which we are all witnesses.
In other words, Peter argues that David couldn’t have been speaking of himself when he wrote those words of Scripture because David’s body decayed in a tomb. He then contrasts David’s death with Jesus’ death and physical resurrection to show that the words of the Psalm are describing Jesus, and therefore Jesus is the Messiah they’ve been waiting for.
Acts 13:34–37, the words of Paul according to Luke, is also explicit:
As for the fact that [God] raised [Jesus] up from the dead, no longer to return to decay, He has spoken in this way: “I will give you the holy and sure blessings of David.” Therefore He also says in another Psalm, “You will not allow Your Holy One to undergo decay.” For David, after he had served the purpose of God in his own generation, fell asleep, and was laid among his fathers and underwent decay; but He whom God raised did not undergo decay.
There’s no mistaking that Paul is contrasting a body decaying in a tomb after death with a body that, because it was raised to life after death, experienced no decay. These are very physical arguments that Paul is using to prove his claim that the blessings God promised are given through Christ.
The charge is sometimes made that Paul knew nothing of a physical resurrection, but instead, the idea that Jesus was physically resurrected was invented later by non-eyewitness Gospel writers. This objection is intended to show that the earliest Christian tradition was of a spiritual resurrection of Jesus.
However, Acts was written by Luke, and this has two implications:
First, Luke informs his readers that he expects his words to be interpreted as history. That is, his intention was to record the words and actions actually said and done by the disciples:
Inasmuch as many have undertaken to compile an account of the things accomplished among us, just as they were handed down to us by those who from the beginning were eyewitnesses and servants of the word, it seemed fitting for me as well, having investigated everything carefully from the beginning, to write it out for you in consecutive order...so that you may know the exact truth about the things you have been taught.
(To answer the question of whether or not such an investigation would have been possible for Luke, see this brief argument for the early dating of Luke/Acts.)
Second, Luke worked closely with Paul, joining him in his missionary journeys, and quite possibly became a Christian as a result of Paul’s ministry. Because of this, it’s highly likely that their views on this topic would have agreed.
So keeping in mind that Luke placed the words of the disciples quoted in Acts in the genre of history (claiming he was attempting to accurately write what the disciples preached), and that he was closely connected to Paul (and so probably had the same view of the resurrection), and that Paul and the leaders in Jerusalem shared the same gospel (indicating the physical view was shared by them, as well), these two passages point strongly to the disciples’ early belief in a physical resurrection.