Rum Village Park – If you go past the main entrance and turn South on S Gertrude St, there are a series of mountain bike trails on the right side. You should park in the main park and bike over to the trails.
Potato Creek State Park – I haven’t been on the biking trails yet, but I hear they are pretty fun.
Bendix Woods County Park – I also have not yet explored these trails, but the park is not too far out of town and probably worth trying out.
South Bend Urban MB Trail – This is different than the more natural trails, but if you don’t have time to get out of the city, its worth a try.
If you really like mountain biking, check out NIMBA, the Northern Indiana Mountain Biking Association (http://nimba-bike.org/). They go out pretty frequently and are responsible for maintaining all of the trails in the area. They have much more information on places to ride.
East Bank and Riverside Trails – These two paved trails will get you from one side of South Bend to the other. There are great scenic views of the river the entire way and it passes through many different neighborhoods.
Maple City Greenway – This one can get you from Elkhart to Goshen.
Pumpkinvine Nature Trail – You can continue from Goshen on to Middleburry and Shipshewanna, straight through Amish country and past some wonderful bakeries.
Calumet Trail – A nice ride over by Indiana Dunes.
For a more complete list of trails in the state, check out http://www.indianatrails.com/.
All you need is a bike and a helmet. If you aren’t extremely serious (like me) craiglist is the best place to get good deals. If you’re looking for more comfort and performance, you’ll have to do you’re own homework.