I'm on St. Simons as we speak.
The island has a mup that covers most everywhere. It's fine for exploring, but kinda rough on thin tires and too crowded and twisty for really hammering.
The island is not used to bikes off of the path (and traffic is pretty thick on the island), but north of Demre Road it opens up and north of Sea Island road it really opens up. However, if you just wish to tool about you won't need a car the entire time you are here.
There is a bike path between Jekyll and St. Simons but the bridge can be daunting.
Jekyll has miles of bike paths and is deserted compared to St. Simons.
In the Village (the shops around the pier) there is an outfit that does kayak tours daily and they sell kayak route maps. They use the creek on East Bridge road as the put-in. Parking is allowed right there. There is also kayaking around Jekyll and they have more put-ins (but I don't know kayaking on the island well enough to delve to deeply).
Feel free to PM me if you wish.