Neighborhoods I would recommend for you to visit:

Santa Monica (especially the Third Street Promenade & the Santa Monica Pier)

Venice Beach

Farmer's Market/The Grove shopping center (Fairfax and Third)

Old Pasadena (along Colorado Boulevard)

Burbank (San Fernando Road)

Los Angeles Art Museum (on Wilshire Boulevard)

Westwood (near the UCLA campus)

Universal Studios

Disneyland/California Adventure (Anaheim)

Knott's Berry Farm (Buena Park)

Gene Autry Western Museum/L.A. Zoo (Griffith Park)

Downtown Glendale

Hollywood/Highland shopping center

Southwest Museum (you might have to locate it via the Internet)

Redondo Beach Pier

San Pedro Pier

Queen Mary (Long Beach)

Drive along the Pacific Coast Highway (going north or south)

The above are a few sights you might want to check out.