I moved back to Michigan after doing med school out of state last year. I think the rule is 30 days of becoming a permanent resident of MI, you have to change your stuff.
That being said, I wonder if being here for 3 years for training qualifies as "permanent". But if you are a temp resident of MI earning wages here, but stay a resident of IL, taxes get a little annoying.
Registration in MIchigan is pretty cheap. There's no excise tax or anything like that from out of state. Just $15 or so to transfer title (yes, they will make you do this) and around $100 for registration, unless you have a brand new or very expensive car. My 96 Lumina was $60 for reg, our 05 Accord was a little over 100.
License is cheap too, less than $20.
Insurance is not cheap. Everywhere I have lived, it is more expensive here. You should change it, because if you get in an accident, Michigan is "no fault" insurance, which is totally different than everywhere else. If you're in Detroit, your insurance will be more expensive. With one accident in the last three years, I paid $600 every 6 months for an 05 Accord, with broad form and comp with $100 deductibles. This was about $400 a year more than I paid when I lived out east. For dirt insurance on a 1996 Lumina, it's about $250 every 6 months, but the accident isn't on there anymore. And I'm old, so I get good rates
Around here, people tend to go with Allstate and State Farm. I've called Geico, Progressive, LA, State Farm, and Allstate, and allstate was the best for me.