I've lived in South Bend for just over 3 years and have found South Bend itself to be pretty bicycle unfriendly. However, if you can afford to live up around Granger, you are close to some good roads that lead up into Michigan. Same with Niles, Michigan, just 8 miles north of Notre Dame. There is also good road riding to the south of town with OK access to the country road system.
I live in the heart of South Bend near IU South Bend. I bike a lot around town and have often been . . . shall we say . . . criticized by South Bend drivers. There is a push for more bike lane striping throughout town (newly re-elected mayor promised 10 miles a year for the next 5 years), but it's being met with some resistance by council members.
Outside of town, places like North Liberty, Osceola, and New Carlisle are all within 12 miles of SB and would give you good access to paved county roads. North, check out Niles, Cassopolis, Dowagiac, Three Oaks, and Galien, Michigan. These are all within 30 miles.
Once you get here, check out Outpost Sports for Trek and LeMond bikes and good mechanics (they also do a summertime Tuesday night ride out of Granger with several levels of riders (including the "Turbos" who do 50+ miles in 2 hours)). ProForm is the high end bike shop with good mechanics. Albrights is the "family" bike shop carrying with Treks, Giants, and Schwinns.
I make a pilgrimage to the "holy land" at least once a year, usually the first weekend in May. Sometimes for a week in June. South Bend is the founding place of Studebaker. Vestiges of the old factories are still visible and there's the National Museum there. My friends and I race a '53 coupe at Bonneville and hold the land speed records for XO Blown Gas Coupe and XO Blown Fuel Altered Coupe. I've asked about biking around there but never received any responses.