Depends on what you're into. The beach is right there, it's ridiculous how close it is to downtown. People have brought up the Getty, well there's something called The Getty Villa right off PCH. Great views of the ocean, beautiful spot to walk around and soak it all in and you can hit their main campus(it's the size of a moderate theme park, I don't know what to call it) if you're into that sort of thing on the way back. For my money I like smaller, well curated museums.
To be honest, I would get a car and drive around California rather than stick to LA. It's just not like NYC or something, nowhere close. But if you're willing to drive, there's San Diego, the desert, Yosemite, places that you'll remember for the rest of your life.