Some of my favorite spots in LA:
The Bodhi Tree, an awesome pagan/metaphysical bookstore. Across the street is Elixir, a lovely tea shop.
The LA Arboretum
Exposition Park - home to tons of museums, the old Olympic Stadium, and some pretty gardens
The Pleasure Chest
Santa Monica: the Third Street Promenade and the Pier
It's also fun to take a random drive up into some of the canyon roads, gawk at the mansions and the views. Just make sure to take a map!