Here's a list from the Express-News and one more with added descriptions:
San Antonio Bike Rides has maps of routes you can take in and near San Antonio. They also have a blog that allows you to follow the creation and revision of the maps. It isn't a club but can help you decide where to ride.
The Hill Country Bicycling Club emphasizes all levels of riding ability, with almost daily rides and several organized trips a year. They say their rides average 10 to 14 mph and they leave no one behind.
The San Antonio Wheelmen is a membership organization. Paid dues allow access to route maps and discounts at a number of bike shops.
The Cool Cats Cycling Club is a recreational road-riding club and no rider is left behind. They say they go slow, take frequent rest stops and have fun.
Alamo Bicycling is a Meetup.com group, which allows them to offer all the features that website including RSVP, messaging and event feedback.