I didn't see anything but body boards at Vero but I am sure there is something around that you could rent. Down the road is Sebastian Inlet. That is a good surf beach. Depending on when you go, the surf is pretty good for learning both at Vero and at Sabastian Inlet. Surf is best when it is about 2 hours away from being low tide and is still on the way out.
I recommend lessons if you can find some. It is allot of fun but hard that iot looks. Watch the currents, I had to help someone out at Vero last year.
You can also sign-up for surf lessions at Typhoon Lagoon...but I think it is pricy.