Some ideas for Los Angeles:
Santa Monica: Explore the pier, relax on the beaches, jog on the beach trail, shop and walk around downtown Santa Monica/ Third Street Promenade, see the street performers, etc.... Its a nice walkable beach city.
Venice Beach: An interesting place, very different than your typical beach, bohemian vibe, people watching is well.... unique. Lots going on during the weekend especially. Many street vendors and performers like singers, break dancers, magicians, etc... You can drive, catch a Metro or Big Blue Bus, or walk to it from Santa Monica.
The Getty Center: A must visit when in L.A. in my opinion and one of the most popular attractions. This excellent and huge art museum is set up on a hill in the Santa Monica mountains on the Brentwood neighborhood with a panoramic view of the city and ocean. People come here to relax in the beautiful gardens, stroll the awesome art collection, and enjoy the views all for FREE (although $10 if parking a car).
Westwood Village: If you want a bight to eat or just want to stroll around an area lots of college students are, Westwood is the place. It is an urban village at the door step to the University of California Los Angeles so always full of young people or professionals from the office towers along Wilshire Blvd.
L.A. Farmer's Market and the Grove: Located in the Fairfax District, one half is the Farmer's Market full of great foods, snacks,and meals and the other half is the Grove, a mixed use shopping area with apartments on upper floors.
Hancock Park/ La Brea Tar Pits: Well, it is just a tar pit, but some people find it fascinating that there is one in the middle of the city. The Los Angeles County Museum of Art (LACMA) and Peterson Automotive Museum are adjacent here in the Miracle Mile neighborhood.
That is some stuff on the westside. You also could head to Beverly Hills to see a bunch of snobby rich people shop on Rodeo Drive, head to Hollywood, downtown L.A., etc... I tried to give some easy ideas for the westside though since you specifically asked about that.