Chicago has an excellent network of buses and trains, which are collectively called the El (for "elevated," which many of them are). The combination should bring you within ¼ mile of any place you'd like to go. Those accustomed to cities will likely be comfortable on any train, anytime. Others may want to take extra caution after 11 pm. Buses are almost always safe; there are several express buses running from downtown to destinations like the Museum of Science and Industry.
As of this writing, the fare for the bus is $2, the train is $2.25, and a transfer is 25¢ with a Transit Card; if you're paying cash, all rides are $2.25. Travelers may want to get a Ventra ticket at airport CTA stations or any visitor center.
For directions to specific places via public transportation, for public transportation maps, and for places to buy Ventra Cards, see www.transitchicago.com.
If you drive downtown, park in one of the giant city-owned parking lots underneath Millennium Park or by the Museum Campus, which charge a flat fee. Private lots usually cost double.