A car is essential — the city is spread out and most attractions are not walkable from each other. Downtown itself is small enough to cover on foot. The Merritt Island refuge is 15 minutes east across the Max Brewer Bridge; Kennedy Space Center Visitor Complex is 25 minutes south on SR 405 and SR 3.
Limited Space Coast Area Transit (SCAT) bus service runs between Titusville and the rest of Brevard County, but the frequency is low and a rental car is the realistic option for any visit longer than half a day.