Christianity isn't directly based off of any other religion, just to get that out first, at least not in the way you can point to, for example,
modern Wiccan practices being an offshoot of Gardnerian Wicca.
Christianity, though, is heavily influenced by several other religions. In some cases this influence can be seen in the form of religious practices,
and others in the imagery invoked in Christianity's stories.
Judaism, obviously, is perhaps the strongest influence on Christianity, Jesus being a Jew and all
Beyond that, though, Zorastrianism has a number
of myths that are very similar to Christianity. This isn't that surprising as it was a strong religion in the same area around the time of Christ's
life. It has stories of a god being born in mortal form, and an apocalyptic fight at the end of time between the god of good and the god of evil.
Some folks say that Christianity stole these stories, others that the were stories that early Christians would have been familiar with so they used
the familiar themes in a new way to illustrate Christ's teachings.
During the time when Christianity was no longer so persecuted in Rome, it was spread far and wide by the Roman soldiers, as were a number of other
mystery cults at the time. Mithrasism is considered one of the other leading cults at this time, along with Christianity. As has been mentioned, the
timing of Christmas is based off of the celebration of Mithras' birth (December 25th). Early Christianity did this with a number of now-traditional
holidays such as Easter (Esotara). In some cases this was a way of showing the indiginous people in an area "Hey, Christianity is what you're doing
already, we're just giving you the proper name" and in others it's simply that the native people didn't want to give up traditional celebrations,
so they found Christian excuses to have a party that same day
I wish I knew of more primary documentation from the time when Christianity was
expanding throughout Europe, syncreticism is something that fascinates me and I'd love to know more of the syncretic elements that have become part
of traditional Christianity.
 "Stories" doesn't imply that the stories are false or made up, it's just a much warmer word than "accounts" or "claims."