Hi Debbie,
I'm a native Southern Californian and can help you out. You mentioned you will be out here this February so you can do the following outdoorsy things:
skiing: Big Bear mountain in the San Bernadino mountains. This is only a 2 hour drive from LA. I used to go there in the summertime to go hiking, horseback riding, and fishing.
hiking: I recommend you check out
http://www.latrails.com/hike/ for a listing of hiking trails in the LA area. Griffith Park is a wonderful place that is in the Los Angeles area. They have a zoo and an observatory. However, I believe the observatory is still closed for renovations.
You may want to check
http://www.latourist.com/ as well. They have several links to places to visit.
I would suggest going to the Farmer's Market in downtown LA (not far from LAX). Really neat place to go shopping and to eat some good food.
Here are some additional suggestions:
There are some really interesting museums. La Brea Tar Pits are neat to walk through. It's also in downtown LA.
Take a tour of the Queen Mary in Long Beach.
Go to the Santa Monica Pier.
Go to the Venice Beach boardwalk. Only do this in the daylight, though. Here, you will find muscle beach where you'll see lots of big guns!
You can go to Palm Springs if you want to see the desert. The weather's pretty nice out there right now and is only a 2 hour drive from LA.
Hope this provides you with some ideas of things to do. I love Southern California (I now live in San Diego) and I go hiking every weekend!
Lorrayne