San Diego is a great place. I lived there for 10 years, but I'm trying to be objective!
And yes, you can certainly take the train, and that's fairly quick and easy. Not sure how much you may save by not renting a car, though, since there are 4 of you. You may not save any money. Not knowing the cost of a rental car OR the train OR how much you will spend in getting to the train in LA, impossible for me to say. You can probably google the cost of the train, though.
About San Diego:
The Gaslamp Quarters is a good central location. Horton Plaza, which is a different kind of shopping mall is right there. A few blocks away at the most.
Seaport Village is a great little area out by San Diego Bay, with plenty of stores and restaurants and lots of people, especially on a weekend. Technically walking distance (from downtown, i.e. Gaslamp Quarters). About a mile.
Balboa Park, sort of in the opposite direction, also maybe a mile or so, at least to the closest part of Balboa Park. A few very interesting museums (including one aviation museum). Also the San Diego Zoo, which is world famous. The whole family will definitely enjoy this! Much more than a zoo.
Old Town is 3 miles, or so, north of downtown. Very historical area, and some of the best Mexican restaurants that I know of (and I'm not even that crazy about Mexican food, but that was one of our favorite places to go when we lived there).
Mission Beach/Pacific Beach (locally referred to as MB and PB) would be the closest beach from downtown. You could take a bus from downtown. I never took a bus while we lived there, but I know the bus system is pretty good, and distances aren't that great (nothing compared to LA, anyway!). Even if you're not interested in the beach itself (and the water will definitely be cold!), it's an interesting place to visit, and just walk along the boardwalk. Especially on a weekend.
Immediately inland from MB/PB is Mission Bay Park, a VERY large park. If you want to go for a run, this is the place!
A few miles north of MB/PB is La Jolla and La Jolla Cove. Very exclusive area with some great restaurants and shops.
Due west from downtown, across San Diego Bay is Point Loma, a peninsula that stretches about 4 miles south, and ends with Cabrillo National Monument, where you have a great view of the entire city of San Diego as well as North Island Naval Air Station.
You can probably get to all these places with the local bus system, but again I'm questioning the wisdom of this. My guess is that you'll be spending a fair amount of time just getting on and off buses, and if you look into all the various costs involved with public transportation it wouldn't surprise me if the rental car is the less expensive option!
[ Edit: Edited on 21-May-2009, at 00:44 by bgl ]