Getty Art Museum, among world's best art museums, and is the world's richest museum (is free).
UCLA Campus (take the free campus tour with a guide, you'll be blown away, & Westwood Village (free except parking costs a lot, like around $3.00 for every 20 minutes).
See a movie is Westwood, tons of great theaters there all packed in a close area, new movies are usually first shown there before spreading
Drive through Beverly Hills (just east of UCLA, free).
LA Zoo in Griffith Park.
Observatory in Griffith Park.
SRF Lake Shrine where Sunset Blvd meets Pacific Coast Highway (free, magnificent lake & garden place, church but also open free to public, worth seeing, peaceful).
Eat at Gladstones on the beach right by the above.
LA Museums by the Coloseum =several, science, natural history, etc. (inexpensive "donation" admissions + pay parking).
Walk around for an hour at Venice beach = crazy local color (free).
Lots of night life stuff at night around where Sunset Blvd / Santa Monica Blvd / Melrose all meet, just east of UCLA & Beverly Hills (big bucks).
If you like Major League Soccer see if there's anything to attend at Home Depot Center, way down south of LA proper.
That'll keep you entertained 4 days on a small budget, all within a fairly close area.