There are three public transit companies and two primary methods of transportation.
CTA and PACE both run buses. Generally, the CTA covers the city of Chicago and PACE covers the outlying suburbs.
CTA and METRA run train lines. CTA has more comprehensive coverage, especially on the north side of Chicago, but METRA extends farther out into the suburbs and, at times, can be quicker.
METRA has zone based fares. CTA and PACE has flat fares.
CTA passes for unlimited rides are available in a number of packages. They can be purchased in person as well as online.
METRA tickets are also available online.
Google maps route the Chicagoland area by public transit, car and foot. The most efficient way to get from O'Hare or Midway to the Hyatt Lodge is by car, taxicab, or airport shuttle. If you choose to take public transportation to the hotel, you may still need a car ride from the METRA Station.
The closest METRA Stations are listed on MetraRail.com and are all located on the BNSF line.
The City of Chicago also details all of your transportation options from both airports here.
Important information regarding taxicabs:
DO NOT pick up a cab from one of the cab stands at either airport and take it to the hotel. The hotel is in the suburbs and once you cross the city line, you will pay fare and a half for the remaining ride. Instead, contact a suburban taxicab company and have them pick you up to avoid the extra cost. Most suburban cab companies have flat fares to and from the airports.
The following is an abridged list of suburban taxicab companies. Most companies charge a flat fare of $30 or less between O'Hare and Oak Brook and $40 or less from Midway.