The good news is that there are lots of places to fish; the bad news is you need a boat. There are essentially no safe places to wade fish other than in salt water. That said, for $20-$30 you can rent a canoe and fly fish for bass and panfish. About one hour north of Disney World I suggest you check out the Wekiva River State Park where you can rent a canoe and fish for miles but you have to paddle back. Downstream and just out of the park there is Wekiva River Marina. It is mostly a launch site for kayaks and canoes. A word of caution, go during the week. The weekend is full of screamers splashing water, blocking the river, etc. The Wekiva River is a designated as a National Scenic River and is worth the trip as therapy after spending time with the Mouse tourists. BTW the Wekiva River State Park has plenty of trails and a spring for swimming which might entertain the family while you are fishing.
Also check out the Florida State Parks website for there are other places. have fun and I hope you get to experience the real Florida.