What did they claim to see?
The disciples claimed to see Jesus after his death. This is multiply attested. We have the written tradition found in all four canonical gospels independently bear witness to the resurrection. The appearance to Peter is independently attested by Luke and the appearance to the Twelve by Luke and John. We also have independent witnesses to Galilean appearances in Mark, Matthew, and John, as well as to the women in Matthew and John. The Gospels were written as early as 20 years after the death of Jesus and at the latest 40-50 years, according to modern scholarship.
Oral tradition - our earliest evidence
But we have even earlier evidence of the appearances than the gospels. Paul quotes an early Christian creed in 1 Corinthians, which was written in 51 AD:
"For I delivered to you as of first importance what I also received: that Christ died for our sins in accordance with the Scriptures, that he was buried, that he was raised on the third day in accordance with the Scriptures, and that he appeared to Cephas, then to the twelve. Then he appeared to more than five hundred brothers at one time, most of whom are still alive, though some have fallen asleep. Then he appeared to James, then to all the apostles. Last of all...he appeared to me." 1 Cor. 15:3-8
Most historians who study this subject - even those who are very skeptical of Christianity - believe that Paul received this creed either after his conversion or when he stayed with Peter and met with James 3 years after his conversion (See Galatians 1:18-19). This puts us within about 2 to 5 years after the crucifixion when Paul received it, so this creed's origin date is extremely early.
Experts believe it's a creed because of its mnemonic structure and the terms delivered and received. In this creed, we see that Jesus appeared not just to individuals, but groups, including the 12 and 500.
Aside from the creed, we also have more oral tradition found in the summaries of the apostles preaching in The Book of Acts. The resurrection was central to their message. (Acts 1:21-22; 2:22, 24, 32; 10:39-41, 43; 13:30-31, 34, 37; 17:2-3, 30-31; 24:21; 26:22-23)
Paul's own claims
Paul evidently made at least two trips to Jerusalem. According to Galatians 1-2, he interviews the disciples about the gospel, and later he writes that they preach the same message (1 Cor. 15:11). This suggests that the disciples, Paul, and possibly other anonymous Christians were preaching the resurrection from the earliest of times.
The significance of James and Paul
Going back to the oral tradition, it's notable that in the creed is the inclusion of James and Paul. James was Jesus' half-brother. The Gospels say that Jesus' own family thought he was crazy (see Mark 3:21). The Gospel of John records his brothers rejecting him (John 7:5) Yet we know James later became a pillar in the church in Jerusalem, and a martyr for the faith. (Galatians 1:19, Acts 15, Josephus, Antiquities 20:200)
Paul was a zealous Pharisee and recorded himself as persecuting the church before coming to the faith because of an appearance of Jesus. (1 Corinthians 15:9-10, Galatians 1:13-16, Philippians 3:6-7) So we don't just have friendly witnesses, but two people who were staunchly opposed to Jesus who converted based on an appearance. And we know Paul suffered and was martyred for his faith. We'll talk more of the willingness to suffer from the early followers of Jesus at a later point.
From the mouths of skeptics
“That Jesus’ followers (and later Paul) had resurrection experiences is, in my judgment, a fact. What the reality was that gave rise to the experiences I do not know.”
- Dr. Bart D. Ehrman, Author of five New York Times bestsellers. Currently the James A. Gray Distinguished Professor of Religious Studies at the University of North Carolina at Chapel Hill. Bart describes himself as an agnostic.