I'm pretty sure they take you down to Ft. Pierce for the lessons. (which is south of the resort)
My husband and his cousin went over a few months ago... They rented boards in Melbourne - $15/24 hours. Carried the boards back to the resort - and Yes! they fit in the elevator (barely!).
They were out at Sebastian Inlet the next morning by about 6:15. That's where they encountered 3 manatees swimming alongside them!
They then headed back up to Melbourne and surfed in Cocoa Beach for the rest of the day.
He was one "pooped" surfer when he got home that night! He thinks he's 16, or something!
Another great place they tried more recently was on a stretch of beach across from Patrick AFB. No commercial development - just beach. Limited facilities, but a great place to surf ... according to him.
You might want to look at past editions of the Vacation Magic on the member website for the details - or maybe an archived Member Clubhouse.
Oh, and check out
www.cflsurf.com for some great webcams of different spots in the area. He was down surfing at Cocoa Beach pier a month or so ago, and I was able to watch him for about 2 hours from the comfort of my home office - it was awesome!!