I love much of L.A.! I live in the arts district of North Hollywood myself. Los Feliz is quite a nice area. As someone said, you'll be right by Griffith Park where there's lots of nice hiking. Plus the Observatory (highly recommend going there at some point), the L.A. Zoo, the old zoo that you can walk through (not a large area but cool to walk around), train town, and great picnic areas. During October the Haunted hayride is near the old zoo in Griffith Park and a part of the train town is converted to the Ghost Train attraction. Los Feliz also has a nice array of interesting shops and places to eat.
You'll be by one of my most favorite places to eat at as well...House of Pies. Diner food but I love it! Their grilled cheese is good and pie selection is pretty awesome. Birds is also near by (bar/eatery) that's pretty tasty. El Compadre is my favorite Mexican food (they have a location in Silverlake and Hollywood). Toi is my favorite Thai food (they are in Hollywood).
Decent music venues (the Fonda, Club Nokia, Paladium, Hollywood Bowl, Greek Theater, Ace Theater, and more) host some pretty good bands throughout the year.
There are various dance options as well if you're into dance clubs. My favorite is Club Underground in China Town...they do 80s and 90s nights on Friday nights. But if that music isn't to your liking there are many other options. Bootie (mash-up music), Bar Sinister (Goth/industrial music), the exchange (mixed bag depending on the DJs but can be trance, electronica, hip hop, house), and more!
Tons of museums (Natural History, Space and Science center, LACMA, The Getty, The Getty Villa, and so on).
Someone else mentioned the beaches...Santa Monica and Venice are nice to visit for sure. Lots to see and do in those communities. But I much prefer the beaches further north (Malibu on up through Ventura County) myself as I find them prettier and better to walk along but that's just my personal preference.
Public transportation is better than it used to be but won't be as quick as other large cities since L.A. is so spread apart.
Ok, I'll stop now. L.A. is an interesting city/county. It's bigger than what most people imagine it to be and is full of interesting people and sites!
-SideshowAlex