You have to use common sense, like anywhere else. The subway (they call it the "el" even if it's underground at some points) is fine. You can get a week-long pass for a reasonable cost at the subway entrance at O'Hare. This will be good for the buses too. And bus service is pretty good.
People always say "watch out for the South Side" --- that's way too simple an answer. For example, Sox Park and Chinatown are on the South Side, along with a lot of really fine music venues. The Pilsen neighborhood has great Mexican restaurants and art, but getting there by public transit is not recommended for a tourist. Take a cab there and back.
There are definitely bad areas. Englewood and Roseland are not known for their night life or tourist attractions. Hyde Park - near the Univerity of Chicago has lots of interesting things to see, like the Museum of Science and Industry and the Oriental Institute (Middle East Museum). But you have to have your wits about you for Hyde Park at night. And I wouldn't really hang out at Uptown on the north-side at night either unless I knew the area. And I would watch out for areas on the West Side too - although there are not many tourist attractions there.
Nobody talks about bad areas in Paris or London but I can tell you from experience that there are some. So this is not about Chicago as such or even the USA.
If you go to places in your tour guide at reasonable hours, you will be OK. And if you are in an area where you are concerned, go back to your hotel by cab just to feel safe --you're on vacation.