There's also the megabus, which goes directly from Ann Arbor to Chicago--I think it's about the same travelling time as the amtrak (4 hours or something like that), and it can be as low as 20$ one way.
I would recommend against a car. Parking is a pain in the neck and the city seems to enjoy writing parking tickets.
Main campus is in town. North Campus is a short UM bus ride away (5 minutes).
Megabus can be picked up at a Park & Ride (also serviced by free UM shuttles I believe) on the south side of town. The bus runs to Union Station in Chicago.
Amtrak has an Ann Arbor station located on the north side of town within walking distance to off campus student housing. I wouldn't want to walk it with a lot of luggage but there are usually taxis around the station when the trains come in. I also wouldn't be surprised if the shuttle buses ran down there as well. The train also runs to Union Station Chicago. Though it being Amtrak, you trip can take anywhere from 4 to 14 hours.
Detroit Metro Airport is about 30 miles away and is serviced by many pay shuttle services.
The train station is closest to campus, though the airport shuttles can probably drop you right onto campus as well. If you're flying into Chicago it may be only slightly more expensive to add the leg to Detroit.
The train is usually the most convenient. vinceh's info is pretty good. 14 hours would be highly unlikely. It usually ends up being between 5 and 6 hours. The train is usually late.
You can also look for a ride. There are a lot of U-M students from the Chicago area going back and forth especially during breaks.
You definitely do not need a car in Ann Arbor so I would not look into getting one. If you have one I would still consider leaving it home for the first year because there are not any good options for parking. After the first year you can decide if there is a convenient place to keep it. It is definitely a luxury but it was nice not having to walk outside when it was below -20.