My CCW instructor referred to the "Peaceable Journey Act." In quick summary, this is a federal law which allows any gun owner to transport a firearm, any firearm, thorugh any state. The federal law trumps any state law. However, the law is designed to allow you to continue on your way through a state, not to stop in that state. So if you have to stop overnight in a gun-unfriendly state (like Illinois), I'm not sure if you might have problems from local law enforcement should they find a gun in your hotel room for example.
In order to be covered under the Peaceable Journey Act, it's my understanding that you need to have your firearm unloaded and locked in a container away from any ammo. Your glovebox and center console do not count, which doesn't make any sense, but it's the law. You then need to have any ammo locked in a separate container somewhere away from your firearm. If you follow these rules, and tell any law enforcement that you are just travelling through, it's my understanding that you shouldn't have any problems.
I live in St. Loiuis and travel through Illinois a lot. Any time I get close to the Illinois border, I have to stop and follow these procedures. I don't know the exact federal statute, but it looks like it's been offered in this tread already. I hope that helps.