As post #7 mentions, Jane Addams would be a good topic to begin with, and her memoir Twenty Years at Hull House is a classic. Another book from the era is Upton Sinclair's The Jungle, which falls more or less in your time period. Researching the Haymarket Riot would give you a wonderful picture of the various radical elements on the ground in Gilded Age Chicago. Death in the Haymarket by James Green is a recent popular history book that deals with that issue. And I'd second (or third) Devil in the White City. As you can tell, much was going on in Chicago during this period, and any survey of the Gilded Age will probably feature Chicago.
For information on this time period, within the period called the Gilded Age generally said to span from 1865-1900, you might research some specific movements. You could look up Settlement Houses, such as Jane Addams' Hull House. You might look up the history of the Union Movement from the pro-Marxist National Labor Reform Party of 1872 to the United Mines Workers Union strike of 1902.
Can I just support the suggestion made in #3? Devil in the White City is an excellent book and a very exciting read, and certainly taught me a lot about this period of history and location. Although it is strictly fiction, it is excellently researched and will do a lot to add to your knowledge of the colour and flavour of this period of history.
Numerous sources on this topic can be found by searching Google Books. Note, in the link below, how your key terms have been highlighted in bold to make finding information much easier. If you click on a link for a particular book, you will often find that the search term is discussed on many pages within a book, not just on the page to which you are immediately taken. You can then ask Google Books to arrange the relevant pages either in numerical order or in order of importance. Google Books is an enormously valuable resource for any kind of research.
Doing some research on the "muckrakers" of this time period will provide you with history on the crime syndicates, the unions, etc that were forged in Chicago. [see the history of the Chicago Stockyards and check out Upton Sinclair's muckracking realism in his book, The Jungle which exposed the terrible conditions in which men worked in the stockyards.] Chicago has a long history of corruption, like most big cities.
Here are some online articles you may find useful.
This encyclopedia entry discusses the industrial revolution in Chicago and how it shaped the working environment and population.
This article talks about the development of the automobile and America's first auto race, held in Chicago in 1895.
This is a great question. The first step in any good research project is to know what is out there in terms of sources. I would also say that one source is not the best approach. To get past the blind spots and biases of all sources, you will need multiple sources. Here is a basic way to start your research.
First, you can go through the archives of newspapers in the library. For example the first edition of the Chicago Tribune was in 1847. So, you can read the headlines between 1870-1900 to get a relatively quick overview of what the times were like. Most likely, this material is also digitized. So, you can do quicker and more efficient searches.
Second, I would try to read a few books on what Chicago was like from that time period. At a library, do some searches about Chicago and look at publication dates.
Third, go through online databases and look through journals. Universities have subscriptions to databases like JSTOR, which has hundreds of journals. Key word searches can do wonders. For example, look for topics like crime, poverty, wealth, and the like. You will get lots of information.