Here's my two cents about group riding, although others may disagree with me. Before you do a group ride or an event ride in particular, get comfortable riding on streets and roads either on your own or with a small group. Learn how to use hand signals appropriately and know the rules of the road as they relate to cyclists.
Once you feel relatively comfortable, I personally would recommend that you do an organized ride with CIBA (the Central Indiana Bicycle Association). They are offered most weekend days in and around Indianapolis. If you go to www.cibaride.org and look for the "non-repeating ride" schedule, you'll find a list of the rides, along with contact information for each ride's leader. Feel free to email the ride leader and indicate that you'd like to do the ride and would appreciate being able to ride with a mentor if at all possible. A mentor will help show you the ropes of riding in a group. Then introduce yourself to the ride leader the day of the ride. Even if you don't have an official mentor, someone will likely be available to help you out.
I've done the Hope Ride. It's fun as others have suggested, but I think it could be a bit overwhelming for someone new to riding. While the crowd does eventually thin out, it's pretty thick near the beginning and thick with other new or casual riders who often ride a bit erratically (nothing against new riders, but I think there's a tendency at casual event rides to sort of forget that they're on a public road. With cars and other users).