Metra Electric trains stop at the Pullman station (111th Street). Please take a look at the www.pullmanil.org website for detailed directions on how to get from the train to the Visitors Center. If you head in the wrong direction, the surrounding neighborhoods can be pretty rough.
The trip from downtown takes less than thirty minutes by train. See the attached website link for schedule information.
Metra Electric trains can be picked up at the Randolph Street station (at Michigan Avenue) or Van Buren Street Station (at Michigan Avenue). Pullman is listed on the schedule as a flag stop so you must tell the conductor that you want to get off at Pullman.
The South Shore (NICTD) trains stop at 115th/Kensington which is a little further of a walk. The South Shore trains also run from the same two stations, be sure to check the schedule as not all trains stop at 115th/Kensington.
CTA runs buses on route #111-Pullman that runs along Cottage Grove Avenue and stops at 112th street, right across from the Visitor Center (copied from www.pullmanil.org website).
See the attached website for route map and schedules.
Personally I would take the train over the bus.
The best place to start is the Visitor Center.
Coming from downtown Chicago, take I-94 East to the 111th Street exit (#66A). Go west on 111th St. to Cottage Grove Ave., turn left. Go one block south to 112th St., the Visitor Center is on the left, on the northwest corner of 112th and Cottage Grove. There is a parking lot at the center. The Visitor Center has a mural on it's north wall.
From I-94 (the Bishop Ford) exit at 11st Street and head west. Turn left at Cottage Grove Avenue. Parking is usually available at the Visitors Center and in front of the Hotel Florence. The neighborhood is best explored by foot.
As this was a railroad town, I think it's quite appropriate to reach Pullman by train. The Metra commuter system stops at 111th/Pullman and 115th/Kensington serve the neighborhood. The South Shore line (service to and from northern Indiana) stops at 115th/Kensington. Exit the stations and walk east.