For the most part, all the beaches are on the gulf. However there are a handful of beaches along Tampa Bay that are closer, but not as nice. They are both pretty much the same temperature. The main difference is that the beaches along the gulf are what Florida is known for
nice resorts, sand, water, etc. The beaches along the bay are usually part of a city park so its mostly just local people. Also, the bay is surrounded by land on 3 sides, versus the gulf where you can look out and see water for hundreds of miles (amazing sunsets!).
If you want to go with the closer option along the bay, its probably just a 20 minute drive from Busch Gardens. Check out the Ben T. Davis beach in Tampa. Its right next to the Causeway, so its a pretty safe location. Theres also a Bahama Breeze restaurant nearby if you are hungry. For what you are looking for (toes in the sand and water, etc.), I think it will do just fine. Plenty of parking available too...
If you get out of Busch Gardens a little early and what to spend some time along the gulf, I recommend you check out Johns Pass Village in Madeira Beach. You can check out the beach, then walk across the street and checkout the boardwalk area, which is alongside the intercoastal waterway. Tons of cool shops and great restaurants (especially if you like seafood).