First off make sure you apply for your FOID card as soon as you move here.
Private Ranges:
Aurora Sportsman's Club - is my personal choice. They have great setup for shotgun, and rifle shooting.
Downers Grove Sportsman's Club - is a great place for trap, skeet, and sporting clays.
Public Ranges:
Buffalo Shooting Range - is really a great place for rifle and pistol shooters. They have some clay shooting but it's all done manually and you have to bring your own clays.
Seneca Hunt Club - is a private club, but they have a great Sporting clays course that is open to the public. They also have 5-stand which is really nothing special.
Mega Sports - is probably you best option for indoor pistol range. I don't shoot pistols, so this is just a shot in the dark, but it is the closest range to Romeoville.
Also all of these ranges have websites that you can check out.