Public bus and rail transportation in chicago is operated by Chicago Transit Authority (CTA).
CTA offers many valuable information on "How to ride CTA", like... what it costs, what the different tickets and passes and cards are, where to get them and so on. We don't repeat this information here because the most up-to-date info can be found oon CTA's website, see links below.
Chicago Transit Authority
Several maps showing the train and bus lines including information about how to use the public transportation system.
Chicago Transit Authority: bus & train system map (PDF)
CTA offers the so-called 'Chicago Card' which is a re-chargable farecard.
Chicago Card (by Chicago Transit Authority)