"He grew up around cars that would later become classics."

Yes and yes—this is an example of what is called the sequence of tenses, which relates the tense of a subordinate clause to that of the head clause. It's the same reason why one would usually write he grew up with cars that were painted blue instead of he grew up with cars that are painted blue. In particular, the use of would here is a relic of its origin as the praeterite of will; similarly, one would use in such a case should for shall, could for can, and might for may.

The above sentence, as I see it, means he grew up around cars which were going to become classics in the future. Is that correct?

Yes.

If yes, does it also mean that the cars actually became classics later?

Yes; would here is concrete and definite.

See the Thesaurus entry from the Macmillan Dictionary:

would modal verb
1 b used for talking about something that was going to happen after a particular point in the past

e.g. Here she met the man who would one day become her husband.

The latter page also lists a number of meanings and usage variants of would. It records towards the end, almost as a second thought, that would is ", incidentally, " used as the past tense of will!