I have to put in a mention for Kurt Vonnegut - he was branded a Sci-Fi author early on, but he didn't really set out to write science fiction; he was more interested in ideas about society and people, and the crazy ways people have devised to kill each other - his WW2 experiences polarized his opinions on this, which is why Slaughterhouse-Five would be a good place to start.
He has a fairly informal, irreverent style, often writing like he's telling a personal anecdote to a rapt audience: unclear plot, placing himself into the narrative, foretelling events, outlining up front what will happen to a character later on.
He often makes narrative excursions to other worlds, not to inject science-fiction, but to ask "what if society were like this instead?", and you can see some of his influence in Douglas Adams' work later.
Breakfast of Champions features a story-within-a-story called Plague on Wheels which may remind you of the Vl'hurg / G'Gugvuntt / swallowed-by-a-small-dog incident in HHGTTG.
You could dip your toes in with his short story collections, but his style really comes through in his novels and essays (A Man Without A Country is one good collection).