Seems a bit silly that I became an armed criminal just because I had to drive through the Peoples Republic of Illinois. This best advice I can give is to learn each state's firearms laws that you travel through. Of course I am told that some people just put the loaded pistol under the seat and drive the speed limit.
The Federal Peaceable Journey laws allow you to legally transport a firearm through a state where you are not licensed provided you are legal at both the start and the end of your journey. There are some other things you must do to be compliant.