If you're going to hire a plane & fly yourself then the closest aerodrome is Merritt Island (KCOI). Hire car is available or use of the courtesy car for a limited period (~2hrs is generally considered a reasonable maximum. It's polite to return the car with a full tank).
Merritt Is. is the same island that has the space centre &the aerodrome is ~15-20nm from the launch sites. All flying ops close during shuttle launches & recoveries due to the TFR that is promulgated. Very simple drive from the airstrip to the space centre. Turn right leaving the airstrip onto SR3 (Courtenay Pkwy) & follow the road until you bump into the space centre.
Otherwise I95 is about the quickest to get from Ft Lauderdale to the Space Coast if driving. Take the SR408 (the 'Beeline') exit East to cross the Indian River then left onto SR3 (Courtenay Pkwy). Follow the signs to the space centre.
I sat on the hangar roof at work at KCOI to watch the shuttle launch the other week.
