If the Tanach (Jewish Bible) doesn’t mention heaven or hell, where did the Christians get this idea from?

There’s more to Judaism than the Tanach.

As discussed in Bosom of Abraham and Paradise, the notion of heaven as a place where the righteous dead go, rather than Sheol for everyone, is a notion that was kicking around in late Judaism, including Jewish papyri and apocrypha such as 4 Maccabees.

That understanding of heaven is mentioned a couple of times in the Gospels, and was developed further by early Christian writers; but it was not alien to the Judaism of Jesus’ time.

Ditto hell: Gehenna as an antecedent to Hell is not much in evidence in the apocrypha, but it’s there in the Targums and the Talmud.

(I gotta say, btw, this was the result of 5 minutes on Wikipedia, and I’m astonished that none of the answers given looked into Jewish antecedents to heaven and hell.)

See also:

• Are there Jewish equivalents of heaven and hell?
• Do Jewish people believe in heaven or hell?