Right by my apartment, I can't help but come here often. It's situated right around the historic La Brea Tar Pits. One needs to spend an entire day here. Also love their magazine/book store.
If you make it to the top, you can get an incredible 360 degree view of practically all of Los Angeles. Below you to the South is Hollywood, and beyond that, on a clear day, you can see over a hundred miles of mini malls all the way to Mexico. To the West on your right go the hills until they turn into mountains and meet the Pacific Ocean. Behind you to the North stretches the entire San Fernando Valley, above which stand the gorgeous San Gabriel mountains. To the East the mountains eventually disappear above hundreds of miles of urban sprawl, and beyond that is the dessert.
The LA Opera at the Dorothy Chandler Pavilion on Grand Street in downtown LA is a must during the fall and holiday season when they have a solid lineup of opera and ballet. Very inspiring.