[past event],后来 + [clause]
[subject] + 后来 + [verb phrase]
后来 means "afterwards" or "later on". It looks back at the past: something happened, and some time after that, something else did. It stands on its own at the head of the clause, or right after the subject.
The gap can be short or years long. 后来 is the word for moving a story forward.
后来 can also go before a noun with 的: 后来的事 is "what happened later".
后来 can't point to the future. For "later" or "from now on" use 以后.
后来 doesn't attach to a verb or a time the way 以后 does. "After class" is 下课以后.
然后 is the plain "and then" for the next step in a sequence, past or future. 后来 adds that time has passed in between.