Hi, you could use either as part of a sentence: ‘She cared after (or for) several generations/generation after generation of the family’ What sounds better depends on what you want the emphasis to be. The second makes it sound like a very long term commitment - ‘generation after generation’ is stronger than ‘several generations’ in emphasising the length of time involved. The first sounds more natural however, less like hyperbole.

The correct expression is "cared for". There's no such phrasal verb as "care after".