It depends on what you want. And also, your specific question of where to go AFTER a Disney vacation is different from those seeking day trips.
You have two choices -- East Coast (Atlantic Ocean) or West Coast (Gulf of Mexico). The beaches both places are great, but different.
On the Atlantic side, you will find "real beaches" with real waves, surf, etc. Many places on the Atlantic side tend to be crowded and touristy (Cocoa Beach, and especially Daytona Beach), but there are some really beautiful beaches which are much more relaxing (Canaveral National Seashore, St. Augustine, etc.)
The Gulf side is more tranquil and might be a better choice if you have little kids who might be intimidated by the waves of the Atlantic. A few areas on the Gulf are pretty touristy -- anywhere around St. Pete Beach, and also Ft. Myers Beach -- but generally the Gulf Coast is calmer.
There are a couple of very cool places, depending on which way you go.
On the Atlantic side, St. Augustine is the oldest continuously inhabited city in North America -- founded in 1565, 55 years before the Pilgrims on the Mayflower, and 42 years before Jamestown. Very cool, very historic and also an interesting town with some great beaches.
Most of the better places on the Gulf side are either further south from Tampa (Marco Island, Sanibel, Longboat Key, etc) , or way west in the Florida Panhandle. But just above Clearwater (which has a nice beach) is Tarpon Springs -- a Greek sponge-fishing town with a very interesting history and culture.
Do some research on non-Disney tourist sites to get a better idea what's out there.
Good luck, and have a great trip!