Rent a car if you can. There are cheap places like Rent-a-Wreck. LA does not work without a car, and if you've never been to Los Angeles you will not want to miss out. LA is a car town, it is nothing like the East Coast cities.
Once you have the car, go see the Beach towns (Venice, Santa Monica, Malibu) and places like Hollywood, the Sunset Strip and downtown LA. You can take the car out to the Desert (go to Joshua Tree National Park) or into the beautiful wine country behind Santa Barbara.
Greater LA has virtually every type of city or country available - go shoot all of it.