First, I'd recommend against trying to use cabs in LA.... It will cost you a small fortune and they are much harder to find outside the airport and certain downtown locations.

Renting a car is probably your best bet if you are flying into LAX. While many of the airport (and a few west-side) hotels have free pickup at LAX, those closer to Hollywood & Burbank almost certainly do not....and will just refer you to car or van services. Some of the cheaper car rental agencies tend to have sub-$20 a day specials (Advantage, Fox, etc.) in off-season like march.

You might consider flying into BUR, which is closer to all of the sites you mention than LAX. It is marginally possible to use public transit if you are flying into Burbank and you are not hevily burdened with luggage... (see http://www.mta.net/) Car rental options are not as diverse, and you are not apt to find any real bargains....but the same prices as you'd expect from the majors at LAX.

Weather in March? Usually pretty good. The temp. usually runs in the low 70s and upper 60s, with nights in the lower 60s and 50s. It's technically the "rainy season" although that's only about 3" in the entire month. You could see days in the high 80s...or the occasional night near to freezing, although those are rare. Classic local weather discription on the Hollywood side of the hills: "late night and early morning low clouds and fog burning off to a sunny & mild day." I think most of the weathermen have that taped...