You can (usually) legally transport a firearm across state lines if:
-You can legally possess said firearm in both states
-It is unloaded and secured in a locked container
The law gets a little fuzzy when you stop somewhere though. If at all possible don't get a hotel/motel room in an anti state.
Check on Michigan's firearm laws and the other state(s) you will be passing thru. I think (don't trust my thoughts) that Michigan, Ohio and Indiana are fairly firearm friendly. And I know that Illinois's laws reflect Chicago (not firearm friendly).