Beyond the usual "The Getty is great, LACMA is pretty cool, some of the old classic movie houses are palaces, yeah the traffic is total butt" stuff?
Go see a comedy show or podcast show at Largo. It's a great venue.
Go to the Museum of Jurassic Technology (very, very hard to explain)
If that isn't your speed, check out the Velveteria (yup, a museum of velvet paintings)
Amoeba Music is one of the greatest record stores in the world
If you're there on the weekend and want a workout, check out the pick-up beach volleyball games on the beaches a few blocks south of the Santa Monica pier, then walk a block to Cha-Cha Chicken for a bite afterwards
Iam8bit has their gallery in LA so check to see if their current show interests you. GiantRobot has a big store in LA too if you're into figures and whatnot.
Head to Koreatown for some amazingly good (and cheap) bbq.