If you are already at Union Station, why not just take METRA's North Central Line to O'Hare Transfer.

Schedules here:

http://www.metrarail.com/Sched/ncs/ncs_wkout.shtml

Fare is US$3.80 one way, half that for Seniors.

At the O'Hare Transfer station, a bus (low floor, IIRC) will meet you and take you to the end (Remote Parking Station) of the Inter-Terminal people mover called "ATS" (or as I like to call it, O'Hare-VAL):

http://en.wikipedia.org/wiki/Airport_Transit_System

Then ride that to your desired terminal.

Cheap and relatively painless!