I was working out in Los Angeles a couple days ago and although I did spot some stars here and there, (Yes, that is Mario Lopez, a.k.a. A.C. Slater from "Saved by the Bell, in case you were wondering) I found the star gazing at the Griffith Observatory to be much more fascinating.
The observatory is located in Griffith Park which sits on top of Mount Hollywood. Walk around the exterior and behold a panoramic view of Los Angeles. You can get an even closer look with one of the many telescopes surrounding the building but whatever you do, don't forget your camera.
You could easily spend your entire time outside and be satisfied with your visit but once inside, I'd recommend seeing a show at the planetarium. It's hypnotic, beautiful, educational, and very dramatic. I'm sure the narrator was an aspiring actor!
After that, make sure to view all of the exhibits. They have several exhibit halls: Wilder Hall of the Eye, Ahmanson Hall of the Sky, W. M. Keck Foundation Central Rotunda, Cosmic Connection, Gunther Depths of Space, and Edge of Space Mezzanine. Also, you don't want to miss the demonstration of the Tesla Coil in the Wilder Hall of the Eye. But if you do, you can always come back another day because another great thing about the Griffith Observatory is that it's FREE!