Bexter, if you're in the mood for a hike, head to Griffith Park (home to the Griffith Observatory, and featured in countless movie and television productions, including Rebel Without a Cause and, more importantly, the Goodbye, Norma Jean episode of QL). There's several mapped out hikes, including a walk to the foot of Mt. Hollywood. There's a fantastic view out over the LA basin, and on a clear day (which in March, is a possibility) you can just see the glimmer of the ocean on the horizon.
http://www.lacity.org/rap/dos/parks/griffithPK/hiking.htm
It's not that far away from Burbank, and you can get to it quite easily using public transport.
There's a ski resort at Mt. Baldy, about 50 miles away from Burbank, and you can just see the snowy peak in the San Gabriel mountains as you pass out of Pasadena towards the Inland Empire region.
You're not going to see snow anywhere other than on a mountain range, but the skiing season runs November through April, so if you don't mind a bit of a journey, then you can see some snow (I just checked the site, and they had a 12"-30" base today). Check out the website here:
http://www.shopbaldy.com/
Oh, as for tremors, I've been to LA about a dozen times now, and I've only once felt a very slight tremor, so don't worry. The closest you'll (hopefully) get to an earthquake is on the Backlot tour at Universal Studios.