1. Huntington Library in Pasadena. Many acres of groomed gardens and landscape plus a fantastic museum of art. Take the tour; there's some history behind Mr. Huntington's extravagance.
2. Old Town Pasadena is nice. Tons of old brick buildings and fascinating architecture plus street photography. It's a "cool hangout". Check out the "Church of God" on Green street - an amazing grounds stuffed with Japanese gardens, old majestic buildings, streams, and so on.
3. I 2nd the Griffith Observatory, esp. if you're a science buff. The LA Zoo's right next door.
4. Many hikes in the mountains. Great time of year for it.
5. If you take a drive 80 miles out the 10 freeway, there's the windmill farm. Behind that is Joshua Tree National Park. There's a scientific anomaly happening with their procreation, where they're flowering exponentially more than ever in recorded history. Some photos here
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6. As mentioned, Hollywood has lots of interesting shows. The Madam somebody's Wax Museum is a nice day, as is Ripply's believe it or not is entertaining. There's a nice magic show if you can get tickets up on the hill behind it, just up from the Chinese Theater.
Further up Hollywood Blvd towards Beverly Hills is a unique happening part of town for having some drinks, seeing some rock music, etc. Melrose District is also good for that, down near Beverly Hills (but without the expensive shops ... just the sidewalk cafe's).
Enjoy the stay. You'll probably see more than most 10 year residents if you try.