My son and I visited the Capitol on a Saturday morning. If you are interested in seeing the place without a lot of people, Saturday is a great time. On Sat they give free tours on the hour - except 12 & 1. Take the free tour! They start at the top and show you the Senate, House, Governor's office, and more.
Many say the Illinois State Capitol is more beautiful than the US Capitol in D.C. It has a beautiful dome, lots of marble, statuary and paintings.
Well worth your time.