In short: yes, the early Buddhist texts teach rebirth: beings are reborn according to their deeds (karma), and the Buddha described recollecting "many kinds of past lives" on the night of his awakening (MN 4). It isn't a soul moving to a new body, since he taught there is no unchanging self. And for people who aren't sure, he offered "four consolations" (AN 3.65): a good life is worth it either way.
What the Buddha said he saw
"I recollected many kinds of past lives. That is: one, two, three, four, five, ten, twenty, thirty, forty, fifty, a hundred, a thousand, a hundred thousand rebirths."
MN 4
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Rebirth without a soul
Unlike the idea of a fixed soul moving from body to body, the Buddha taught not-self: body and mind are changing processes. What continues is a stream of cause and effect, shaped by intentional actions. That's why karma is central: "beings are the owners of their deeds and heir to their deeds" (MN 135). (Karma in Buddhism)
If you aren't sure: the four consolations
The Buddha didn't ask the Kālāmas to accept rebirth on faith. He described what someone with a mind "free of enmity and ill will" gains either way:
"If it turns out there is another world, and good and bad deeds have a result, then—when the body breaks up, after death—I'll be reborn in a good place, a heavenly realm."
"If it turns out there is no other world, and good and bad deeds don't have a result, then in this very life I'll keep myself free of enmity and ill will, untroubled and happy."
AN 3.65, the first two of four consolations · More: the Kālāma Sutta.
The practical point: kindness, honesty and a clear mind are worth it whether or not there is a next life.
Try it
Start with what's good in this life: a 3-minute kindness practice in Practice. More on how the Buddha saw death: real Buddha quotes on death.
Questions people ask
Do Buddhists believe in reincarnation?
The early texts teach rebirth: beings are reborn according to their deeds (karma). It differs from reincarnation of a fixed soul, since the Buddha taught that there is no unchanging self; what continues is a stream of cause and effect.
Do I have to believe in rebirth to practice Buddhism?
The Buddha's 'four consolations' (AN 3.65) show that a life free of ill will is good either way: if there is another world, you'll be reborn well; if there isn't, you live 'untroubled and happy' in this one.