C.K.
Check out the condos on a site called VRBO.com. (Vacation Rentals By Owner). My SIL is from Puerto Rico, so we've travelled there a number of times as an extended family and have had very good experiences with the condo rentals we've gotten through VRBO. You should be able to find a 3-4 bedroom place that holds your family with beach/pool access at a very resonable price. (We've even got a deal once that was the same price as 1 hotel room!)
Definitely rent a car (just like driving in the US, but there is a lot of traffic in San Juan itself), since it's also a great way to see the whole island. With 10 days, you'll have plenty of time to explore the whole area. Most airlines let you check a car/booster seat for little or no additional charge (it's not considered normal baggage), so look into bringing at least 1 or 2 of yours instead of renting them. You may also be able to carry the booster seats onto the plane if they're the small ones.
A car will also allow save quite a bit on accomodations if you follow the coastline towards Carolina. It's much cheaper if you go a little ways outside of the city. From there, you'll be about 20-30 min from Old San Juan (a must see), and about 45 min away from El Yunque Rainforest (another must see). Other good road trips (no more than 2-3 hrs drive) are Arecibo (where the observatory is from the movie Contact), Camuy (really neat caves), and Ponce (incredible beaches). There's also a little island called Vieques that you can get to by passenger ferry from Fajardo. That place is like being on Gilligan's island.
Enjoy your trip!