Double Star is arguably the prototypical Heinlein novel, with most of his later novels falling into that mold to some extent, so start there if you want to see what he's all about. You should definitely read Starship Troopers, if only to better mock the sort of half-wit that's constantly going on about "media literacy". Honestly, I have mixed feelings about Stranger in a Strange Land, but it was insanely influential, so it's an absolute must. And aside from The Moon Is a Harsh Mistress, Methuselah's Children and the various other appearances of Lazarus Long are the one's that best fit the description of "transgressive literary fiction that's also a scifi romance".

As for short stories, pretty much anything in his Future History series is of interest, though a lot of it is obviously dated. Outside of that, my personal favorite is "'—All You Zombies—'", but I'm probably the only person () that would put that at the top. "By His Bootstraps" is sort of the predecessor to that one, so give it a look as well.