Spring Break
The college kids generally prefer the Miami/Ft. Lauderdale area. We do see a few of them here on the Gulf Coast, but in far fewer numbers. I believe it's the same over in Cocoa Beach as well. I've been over there a few times during Spring Break, and it's about as quiet as the Gulf Coast.
If you are a AAA member, you can save more than a few bucks on your tickets to quite a few Orlando attractions (including Disney's), as well as the Kennedy Space Center.
I have been over to the Cape several times. To truly appreciate everything, plan on spending the day there. If you can, take the "Then & Now Tour". That takes you to the area where the earliest launches went up. It also takes you to Pad 34 where the Apollo 1 fire occurred. If you can, sit on the right side of the bus, as the route goes as close to Pad 39A as most people can get.
I have not been to the Astronaut Hall of Fame yet. This used to be a separate attraction, until the franchise holder for the KSC tours brought the previous owner out.