This guide covers LA pretty well: https://sites.google.com/site/theredditguidetola
I definitely recommend going to the Griffith Observatory and/or The Getty for great views of the city(when its not covered by smog).
Beaches in the Malibu area are very nice(Zuma, Leo Carillo). If you want to free dive, there are some nice spots near Laguna Beach.
Hiking is also another option if you are into that type of thing. The hike up to Mount Wilson is very nice. 12 mile round trip, not too bad if you are in moderately fit condition. Views are great from up there.
If you want to go shopping or just check out landmarks, Hollywood and Pasadena are good places for that.
Food is a big thing in LA, you can find pretty much any kind of food you want here. Little Tokyo has some very nice and cheap restaurants. Check out Shin-Sen-Gumi if you are a fan of ramen, its one of my favorites and a great spot to go with friends.
I don't have time to list more stuff right now, but check out the link I posted above!