There are a few different options. As far as stops off the Wolverine, Ann Arbor and Dearborn would make most sense, with Dearborn being the best option if you wanted to take a taxi (though if time were most important, getting off at Ann Arbor and taking a taxi there would be fastest). You can also take the Blue Water to East Lansing and catch Michigan Flyer there, though that would be considerably out of the way..
From Ann Arbor Michigan Flyer has partnered with AATA to offer the AirRide service to DTW (myairride.com) - which is basically the same as the standard Michigan Flyer, though it now stops at the downtown Blake Transit Center (and on some trips the Central Campus Transit Center). As such, you can simply take a downtown-bound AATA bus from the Amtrak station (either #17 right at the station, or #1/#2C from Beakes St) to the BTC and transfer right there to the AirRide. You can also walk from ARB to the BTC, though at 0.8mi that probably wouldn't be advisable in winter or with lots of luggage. Fare for AirRide is $12 each way with reservation, and for the local AATA bus the fare is $1.50. Do make sure you check the schedule before planning a trip - you won't be able to get AirRide or AATA late at night (i.e. coming from #354), and the AATA local bus to BTC doesn't run after 6pm or so on weekends (though AirRide runs a bit later).
From Dearborn, in addition to the taxi there are bus options as well. SMART has some bus stops near the Dearborn station and services DTW as well. However, you'll probably have to make at least one transfer, maybe more if it's a weekend. That would probably be your cheapest option ($2.00 + $.25/transfer), though it would take longer than the AirRide from Ann Arbor. Also, it may be difficult to do this with lots of luggage as it's just a regular city bus that mostly carries commuters (whereas AirRide uses buses more like a Thruway bus/Greyhound).
Coming from the west, I'd probably recommend Ann Arbor if you were going to do a bus since it would be quicker and the transfer would be simpler. However, if you wanted the cheapest option taking SMART from Dearborn would save a bit of money - though it would probably involve the most hassle and be the slowest of the various options. If you wanted to just take a taxi, Dearborn would be cheaper, but Ann Arbor would be quicker from the west given that Dearborn would involve backtracking.
In any case, with an Amtrak-airport transfer like this, I'd make sure I'd allow plenty of time for the connection or have alternatives if you miss your flight - you don't want a 1 hr delay to make you miss a non-refundable, non-exchangeable flight. Taking #350 and catching a later flight from DTW would be safest...