Some of my favorite spots in LA (keep in mind that my wife and I are suckers for weird artifacts, books and collections):
http://www.getty.edu/ - it just doesn't get more civilized than this gorgeous museum overlooking the city. World class collections and it's completely free. Very good free concert series, but you need to get tickets in advance often.
http://www.obsoleteinc.com/ Weirdness. A fantastic store in the Venice area that carries bizarre artifacts ranging from cool industrial and medical equipment to old wooden dolls.
http://www.mjt.org/main2.html - More weirdness. A typical exhibit is a portrait gallery of dogs of the russian space program. We can easily spend a whole afternoon wandering this place.
http://www.thesecretheadquarters.com/ - Amazing little comic book store in the Silver Lake area (which is itself a great area to walk around and explore). Recently ranked as one of the ten best bookstores in the world by The Guardian.