If you want to park right by the Joe, I would recommend parking on Cobo Roof. It is cheaper ($10 instead of $15), and less of a pain to get out of than the arena garage. You will still have a little bit of a wait, but not nearly as much as the garage (this has been my experience, at least).
If you want free parking.. you can park at the Hard Rock Cafe's garage, go into the restaurant, go to the gift shop and have your parking validated. They have a free shuttle going over to the Joe, and they serve free shots on the ride.
I loathe the people mover after games because there are a million people trying to get on and it takes forever. I usually park on Cobo Roof or Hard Rock.
There are some hotels in Troy, which is not too far from downtown, that are nice. Also, Hockeytown Authentics is in Troy, if you've never been there, it is a fun place to visit/shop.