The two main cab companies in Chicago are Checker Cab and Yellow Cab, and you can hire one just by calling T-A-X-I-C-A-B, or 312-829-4222. A taxi ride to or from O'Hare Airport can range anywhere from $30 to $50 and a ride to or from Midway Airport can cost as much as $20. Taxi fare from Downtown Chicago to Hyde Park is about $18.
If you're scheduling a pick-up, make sure to tell them your name and the time you need to be picked up. It's also important to call half an hour before pickup time to make sure that there are no misunderstandings.
Omega Shuttle Service also offers transportation from Hyde Park to Chicago's Midway and O'Hare Airports. You can find fare and scheduling information as well as boarding locations on their website, http://www.omegashuttle.com/.