LACMA, absolutely a fanstastic art gallery located in the heart of downtown. And of course, there is the famous installation of multiple lamp posts that you may have seen in many movies.
From downtown LA, I had also traveled to Long Beach. Yet another quiet and beautiful place. The beach was very nice and is literally a 'long' beach. :-) I got a great tan and indulged in many chapters of a great novel (The Continuity Girl). I tried visiting the art gallery also, but unfortunately it was closed early for a private function. I also wandered over to the aquarium and walked around the marina. There's a nice assortment of restaurants in this area. Across the way is a mall and mini amusement park.
By far my favorite area is Santa Monica. I enjoyed all the unique restaurants, vintage stores, yoga studios and beautiful pier. For the shopaholics there is Santa Monica Place and Third Street Promenade. In the summer weekends there's a nice farmers market, great place to visit if you like fresh produce. If you enjoy vintage shopping a great place to go is Wasteland. They have funky finds there. I proudly support recycling and reducing my carbon footprint. :)