Since I grew up in L.A. and go back often to visit family (having recently used both bus and rental car), I can give my opinion.
Contrary to what has been said, the buses and the museum ARE SAFE. However, L.A. is so spread out that it would take several hours and transfers to get from one place to another. I use public transit in cities all over America, and L.A. is probably the worst in terms of convenience and figuring out where to get on, etc. (However, I have not tried their new light rail). A rental car would be cheaper than a taxi for that distance (although rental cars in L.A. are more expensive than anywhere else in the U.S.).
The Natural History Museum, though safe, is not very good at all. I wouldn't bother. The aquarium, on the other hand, is quite good.
If you like large botanical gardens, either L.A. County Arboretum or Descanso Gardens are quite good. (Arboretum has Caribbean-style house that was the set for 70's/80's tv show Fantasy Island).
The previously mentioned Griffith Observatory is accessed by a bus (they no longer allow cars due to congestion) that departs from a small lot right next to the zoo parking lot. So that would be a good place to go after the zoo closes. The attraction is not so much the stars, but the cool building (featured in several movies) and the stunning view of downtown Los Angeles. As you go up on the bus, notice a short tunnel to your right, just before you reach the parking lot. That was the tunnel that leads to Toon Town in the film "Who Framed Roger Rabbit?"
I have not been to The Getty, but my brother loves it. I have seen it up on a hilltop from the freeway and it looks very cool. You also take a monorail from the parking lot to get there, which is fun. My cousin commutes along the freeway pass underneath it and often sees deer on his way to or from work.