Largest state park in Alabama
Oak Mountain State Park is the largest state park in the state of Alabama. And, since it's only 20 miles south of Birmingham, it's a great place for a family day-trip or simply to escape from city life for a few hours. If you're visiting Birmingham in summer (and can't make it down to the Gulf Coast), I'd suggest heading to Oak Mountain State Park and spending the day at Double Oak lake. There's a swimming area, a beach, and even volley-ball courts. It's not the Caribbean, but it's really very nice taking into account you're in central Alabama!
Two other things you can't miss at Oak Mountain State Park are Peavine Falls and the Treetop Nature Trail. Peavine Falls is a 60-foot tall waterfall that's an absolute blast in summer. Try heading up the trail in the morning before gets crowded and enjoy a picnic and some swimming (the water is icy, watch out!) in the afternoon. The Treetop Nature Trail is a sort of elevated walkway that takes you past various shelters with owls and vultures and other birds in them. The kids will love it!