The first option is to take a taxi. Taxis are available at both airports and a one-way ride will cost about $40 - $60. Taxi stands are outside the baggage claim areas of the terminals. There may be a shared rides option that will allow passengers to save money while traveling from one airport to the other.
Another option is to take a ride share service like uber. Ride share services like uber or Lyft will probably cost somewhat less than a taxi (but confirm pricing before you go, since uber's surge pricing can increase costs). uber started providing pickups and drop offs at both O'Hare and Midway airports in November, 2015.
To get an uber car at O'Hare, business travelers need to head to the departures level upstairs. For Terminal 1 the pickup area is zone 1. For Terminal 2 the pickup area is zone 1 or 2. For Terminal 3 the pickup area is zone 2. At the International Terminal, travelers should head to the Arrivals level (downstairs) for uber cars.
To get an uber at Midway, business travelers should aim for the upstairs, departure level. Exit through the door UL-1 for the right pickup area.
In either case, uber suggests waiting to retrieve your luggage before requesting an Uber ride.
Also note that uberPOOL pickups are not yet available at the Chicago airports. (But you can use uberPOOL to be dropped off at the airports).
But there are other options. Another option is public transportation. Public transportation will work fine and is more economical than a cab ride, but will take some time and requires a transfer.
From O'Hare, travelers should take the Blue Line (lowest level of the parking garage, near elevator 4) to the Clarke and Lake stop. From there, take the Orange Line to Midway. To get from Midway to O'Hare, reverse the process (Orange Line to Clarke and Lake, then the Blue Line to O'Hare).
Another option for business travelers to get from Midway to O'Hare (or vice versa) is a bus or limo service. Coach USA Tri State/United Limo provide regular service between the two airports during the day. The trip typically takes about an hour. Business travelers looking for a bus at O'Hare should head for the Bus/Shuttle Center in the main parking garage. Go to Level 1, near elevator centers three and four.