I'm also in G'ville and at the moment I don't belong to GCC. But I have been riding with a group from UF's College of Veterinary Medicine called Team VetMed. They have been doing a 23 mile Wednesday afternoon ride from the Vet school down 441 to Wacahoota Rd to Willistion Rd and back. Normally they roll at 5:30 PM, but they're trying a 3:30 ride now, but turnout has been sparse. The ride started as a prep for the Horse Farm Hundred, and since that's over attendance dropped. But the early departure time is probably a factor as well.
I have heard of group rides coordinated by bike shops. I'm pretty sure Bikes and More, Chain Reaction and Mr. Goodbike have their own ride schedules. I think even Critical Mass is doing something. For Team VetMed info use Google. Both Century rides of GCC are laid out well and can easily be tailored to fit the distance you'd like.
And since you have a mountain bike the San Felasco Hammock is a great place to ride. It's just south of Alachua. The trails are great as well as the scenery.