Both the Illinois House and Senate have live video and audio feeds of their sessions.  Click below to see real-time video or to find out when the next session will take place:

House of Representatives: live audio/video

Senate: live audio/video

Illinois Legislative Synopsis and Digest (print)

Call number: 328.773 IL63L

Location: Reference [non-circulating]

The subject index in this source is the best way to find bills on an issue that interests you.  The subjects are very general, and make sure to check the subject "New Acts" for all the bills with new content (vs. amending existing law).  The bulk of these volumes is a history of each bill that was introduced: the subject index is in the last volume of the year.

Illinois Legislative Synopsis and Digest (online)

This site contains PDFs of the print Illinois Legislative Synopsis and Digest indexes for several general assemblies, with the subject, sponsor, and other indexes scanned separately to make them easier to download. 

Illinois General Assembly Website

The "committees" link lists the subject focus of each committee and the bills that have been assigned to it, which can be a good way to identify bills within a broad subject area.  The General Assembly site also contains a keyword search box for bills, but each version of a bill turns up as a separate result and the results are quite difficult to interpret.  The most efficient way to use this search feature is to skip it entirely until you have identified a bill number in the subject indexes, then search for the bill text and history online by bill number.