After a young witch or wizard graduates from Hogwarts, they're expected to join the adult wizarding world, and part of that is figuring out which career works for them. During the fifth book in the series, Order of the Phoenix, Harry visits Professor McGonagall for a required career advice meeting, and they discuss which classes Harry should take to qualify for his desired post-grad job (and since Harry wants to be an Auror – a dark wizard catcher – that's basically all of them).
However, aside from becoming an Auror, post-Hogwarts options seem pretty scarce especially according to the adults in Harry's life. It's never revealed what Harry's parents, James and Lily, did for a living, but almost every adult he knows growing up either works for the wizarding government, the Ministry of Magic (which includes Aurors), or teaches at Hogwarts. Though there is a wizarding hospital, St. Mungo's, which is stocked with Healers, and some talented athletes might become Quidditch players, this means there are roughly four viable career paths in the wizarding world, all of which require rigorous schooling and difficult training regimens. The stage play, Harry Potter and the Cursed Child, which serves as a sort of sequel to the original series, seems to confirm the lack of career options — as adults, Harry, Ron and Hermione are all employed by the Ministry, with Hermione even serving a term as the Minister of Magic.