You can take the scenic route or the fast route.
Fast: Highway 401 leads from the port to 528. Head West on 528 until you reach I-95. The South on I-95 to exit 69 (?), aka Fellsmere Road or CR 512. East on Fellsmere, then south on CR 510/90th Ave. 510 will then turn east. Follow to A1A and then turn right on A1A, the resort is just past the turn on the left. About 74 miles.
Scenic: At the intersection of 401 and 528, stay on 401 headed East rather than taking 528 West. You'll shortly see A1A (Astronaut); turn south on it (or you can just stay on 401). This will merge back in to 401 and just keep heading South. 401 turns in to A1A around Cocoa Beach, and just stay on A1A until you get to Vero. About 50 miles or so from the port.