Great timing, I just got back from Jekyll Island a couple of nights ago. I didn't have any experience with the area prior, so it was all new to me, but my family all had a fantastic time there.
We stayed at The Jekyll Island Club and enjoyed it. The staff were all very friendly and they have three different restaurant/bars to eat at without having to drive anywhere. The rooms were a little small, and we didn't have a view from ours, and our air conditioner had two settings: freezing or hot. These were minor complaints though, since we didn't spend too much time in the room. If you're looking for something closer to the beach, the Hampton Inn looked pretty nice, but I didn't stay there, so I don't have any first hand experience with it:
http://www.jekyllislandhamptoninn.com/photo-tour
The beaches were wide open, and the water was warm, but not blue/green. The sand isn't white sand that you'd see in places like Destin, but I actually liked it better since it was really well packed and easy to walk on. I actually saw a guy and his son riding their bicycles on the beach itself without issue.
I didn't play golf this trip, but the course looked pretty nice when we biked past it, lots of people playing. The bike trails are pretty fantastic, and quite impressive. They all look brand new and the vast majority of the trails are covered in the shade of giant oak trees, so we had some really nice rides. We just rented bikes from our hotel, but I saw a lot of people had brought their own.
Electric car rentals are available at the local airport, which is just a couple of minutes bike ride from the Jekyll Island Club. There was a guy in a Waco there giving rides from the airport. The airport seemed to be in great shape, but I didn't see any hangars or fuel available; there were 7 or 8 planes tied down, the biggest being an Aero Commander.
We drove up to Savannah from Jekyll Island on a Saturday to spend the day there as well. It took about an hour and a half to get to downtown, but everyone enjoyed the excursion. We took a horse-drawn carriage tour while in Savannah, which was fairly interesting. It made for a fun day trip off of the island.
If you have any other questions, let me know.