From Downtown to Chinatown

Arrive at Midway Airport, and jump on an 'L' train towards downtown Loop for only $2.25 USD (O'hare Airport makes you pay $5 USD just for leaving from there). You'll ride it for about 25-30 minutes until you get to the Library stop at State and Van Buren Streets.
After walking just 1 block south on State street and 1 block east on Congress, you'll find yourself at HI Hostel Chicago, one of the cheapest and most accessible hostels in the city of Chicago. Drop your bags off; you've made it to Chicago! 
Take a 5-minute walk over to the Pritzker Pavillion on Michigan Avenue, where you're sure to find either free live classical music in the afternoon or a free movie screening of a popular pop culture film in the evening. Enjoy for a little while as you take in Chicago's bustling downtown scene. Check out their online calendar of events to plan accordingly.
Stop at a local drug store like Walgreens or grocery store Trader Joes to grab some cheap lunch snacks to bring along (instead of the expensive restaurants on Michigan Avenue). 
Another 5-10 minute walk across the Chicago River will get you to 400 North Michigan, where you can buy a one-way ticket for $6 to take a scenic, Chicago Water Taxi ride from Magnificent Mile to Chinatown.
A 15-minute walk through Chinatown will get you to Lao Shanghai, one of the best places to get soup dumplings in the city. At 8 dumplings for only $6 USD, you can't go wrong. Enjoy a leisurely lunch or early dinner there before you head back on a train to the downtown loop.
The Red Line Cermak-Chinatown stop is pretty close by, and depending on how tired you are feeling, you could either take the train back to your hostel (Harrison stop) and rest up, or take it up to the Grand Avenue stop and have a few drinks at Second Story Bar, an old-school, cash-only gay bar hidden in the Streeterville district.