G.D.
I'm a relative newcomer to Savannah myself. We've been here about a year and a half. We make it a point to go to most of the First Saturday events on River Street, as they usually have a kids' area with rides, games, and such, especially around holidays. I try to check out local webpages when we're looking for something to do. Try Riverstreetsavannah.com and Savannahnow.com. Also, March is the absolute best month with all of the St. Patrick's activities downtown.
Some other neat places of interest we've found- Lake Mayer has ducks to feed, park and track; Flying Frogs to just go let loose, and the Aquarium out on Skidaway is tiny but interesting enough for my four-year-old daughter! I also want to visit the new Children's wing at Telfair, but just haven't made time for that one yet. Another good one for an easy moms' meet up is the kids play area at the Oglethorpe Mall. It's nice when you have other moms to go and you can just sit around and talk while the kids exert all their energy!