You could go to the Savannah National Wildlife Refuge, located only about 15 minutes from downtown Savannah just over the border in South Carolina.
Savannah NWR Home Page
Savannah NWR FAQ's
Savannah NWR Map
I HIGHLY recommend the Laurel Hill wildlife drive; you can drive along the dikes of an old working rice plantation, and look into the old fields and canals. There's been plenty of alligators and other local wildlife there in the past.
And the best thing? It's completely free, and open seven days a week from dawn until sunset.
Great place for a picnic, too!
And, if you like the Savannah refuge, you can go to Pinckney Island (see
Pinckney Island NWR Home Page), or Harris Neck (very, very cool...old abandoned WWII airfield with lots of nature;
Harris Neck NWR - Home Page).
Also look at the companies that offer kayaking; there's several companies (sorry, but I can't give you the website because of the forum's rules, just google around) that put in at Ebenezer Creek in Effingham County (just north of Savannah), and take you through that blackwater cypress swamp. Very cool. And very close; only about 45 minutes north of downtown Savannah.
Have fun!