I live in Malibu. LA is a very upbeat, chill with your friends kind of city. Going out alone isn't going to be very enjoyable unless you hang out at coffee shops (Urth Cafe is a GREAT spot for this with lots of outdoor seating), usually parking in the streets around there at the one in West Hollywood. You'll probably see celebrities while you're there, that's where they go to read the newspaper. For people watching and good shopping I recommend the Grove of LA and Third Street in Santa Monica. The Getty and the Getty Villa are fantastic. Santa Monica beaches are always busy but there are a few beaches in Malibu that I go to lay out/do homework where I feel fine being by myself. While in LA you should go to the Observatory, hike the Hollywood sign (which I have yet to do after growing up in So Cal). Diddy Riese is a great suggestion; I'd add Millions of Milkshakes in Hollywood. Brunch in the late morning/afternoon on weekends is awesome in LA so google great spots to go, my favorite is Blue Jam. Anything you're looking for in particular?