Depending on how much time you want to spend, I would recommend taking the Orange Line all the way to Midway and/or the Blue Line all the way to O'Hare. Either way you will get a good look at the city and be able to experience the Loop as well. Now I do not know if they kick you off the system at ORD or MDW since they are endpoints; so you may need to purchase mutiple rides. Loop to MDW to ORD and back to Loop should be no more than 3 rides at most. A single ride is $2.25 and a Day Pass is $10. Farecard machines are located right before the turnstyles.
They tell you to get off the train when it reaches the end of the line, but except for the Yellow Line, you don't have to leave the fare-paid area, just stay on the platform and get on the next departing train.
The Yellow Line is an exception because it lets you off on a platform that is outside the fare paid area (boarding passengers enter from a different platform).
That's just for information for anyone wanting to do a joyride. In this case, the OP has a specific destination in mind, so it's not relevant.