Things to see everywhere...
There are unique sights hidden throughout the park. You could spend days googling trip reports and find some very cool spots, but where's the reward in that? Just go, don't be in a hurry, make lots of stops. Drive up towards that mysterious-looking canyon with a bit of greenery around its entrance; you'll find something!
Scotty's Castle is about as touristy as I get, but it is interesting (with petroglyphs as well, not published), and a good jumpoff to Racetrack Valley. Head from the Racetrack down Lippincott Grade into Saline, and camp at the hot springs. Leave Saline via Steel Pass and Eureka Valley, then you can loop back east and south to the main park again, or head north into Nevada.
If you take the route I mentioned, you will have passed several dozen petroglyph sites, ancient rock alignments, fossils, huge quartz crystals, hundreds of mines (many still open, multi-level complexes), a lone bat statue, a marble tub, Charles Manson's "lost world," 700' tall sand dunes, several hot springs, tea kettle junction (bring a tea kettle to leave in the collection), and decently-maintained cabins which you can stay in (if a flag is flying on the flagpole, that means someone is already staying there and wants privacy).
I don't know the southern portion of the park, but understand Goler Wash is interesting, butte valley, Mengle ("Mangle") Pass, and environs. I think there are more cabins down there, too, although being to the south there are more people.
Expect to be buzzed by military aircraft if you enter Saline. It is part of a Military Operating Area, so gets some very cool planes and helicopters.
Avoid the rangers. They don't like when you pick up rocks, and will cite you if your dog is off the leash. You'll often meet recent transfers from east coast national parks, where things are different, more law enforcement oriented. They are so new to the area that they really don't know the best places anyway. A common patrolling practice in the area is for them to set up in a lawn chair on some height, from which they can observe a greater area. The park is huge and there aren't that many of them, so it really is an effective tactic.
Otherwise keep an open attitude about the weirdos you'll see (I might be one of them). There is generally a very mellow attitude once you get off the pavement and away from the NPS crowd. Relax and have fun, don't be in a hurry.
My moment was sitting in a hot spring one night down there, watching the afterburners of jets dog-fighting overhead.
