I think there is a Sunday afternoon ride, but honestly I've never been a part of that. I ride by myself on the weekends.
I've never done the fire department rides with the group. It takes me about 30 minutes to get there from my work, and I can't leave work early enough to get over there and get ready by the time they leave. From what I've heard though, they are a bit smaller.
The Tuesday night ride from capitol park is by far the largest. After the ride the group meets at a local bar/restaurant for dinner and to talk.
If you like to ride by yourself on the weekends, the Hot One Hundred route is a nice route that has 23, 30, 45, 65, 100 mile options. It is a charity ride that has been done for several years and the markings are all clearly still painted on the road. So you don't need any type of cue sheet--just follow the HH signs spray painted on the road. enough people ride this ride that you won't have any problems with cars or people yelling at you (at least I have not).
If you want hills also, the Lake Lurleen loop is very good. 10 mile loop that has great hills you can play with. Not mountains, but good hills. Again bikers always up there, so the locals don't yell/honk at you.
If you do want to hammer it, the Velo City Race team and the University of Alabama Cycle Team are generally a part of the Tuesday night ride. I've seen them as they passed me