There is no train from Chicago's airports directly to South Bend, but you can take the Rapid Transit from O'Hare airport to downtown Chicago, then connect to the South Shore Line which comes to SB airport (about 3 hours). You would then take a cab to campus for about $14 (10-15min taxi ride).
At O'Hare airport, follow signs for "Trains to the city", and take the Blue Line (the only line at the airport) to downtown. Tickets are purchased from machines at the station entrance. Once in downtown Chicago, get off at the Washington station and follow signs to "Randolph/Washington". The South Shore Line's station is on Randolph St. Directions and schedules are at the South Shore website: http://www.mysouthshoreline.com/. (Click "Directions in Chicago".) You buy a ticket at the station or on the train. The South Shore Line trains are relatively infrequent on weekends (several hours or more between trains) but while you wait you will be in quite a nice part of downtown Chicago.
The university is about 2 hours (by car) from O'Hare airport, less from Midway. From O'Hare Airport: Take Route 190 east out of O'Hare to Route 90 east (Kennedy Expressway) toward downtown Chicago where it merges with Route 94 south (Dan Ryan Expressway). Take the Skyway exit off the Dan Ryan and remain on Route 90 to the Indiana Toll Road which eventually merges with Route 80. Get off at Exit 77 (South Bend/Notre Dame).
There is a limousine run by Coach USA (formerly United Limo) which runs from the Chicago airports (serves Midway and O'Hare) to Notre Dame. It will drop you off right at campus. It is about a three hour trip (90 Miles). More info available here.