Parking at the casinos and catching the people mover over is a cool idea. Right after the game the lines to the people mover are huge so when I was in town last, I hung around the Joe and got to take pics, go down to ice level and all that jazz. Killed about 30 mins or so, by that time didn't have to wait long to catch a ride back to greektown. Plus it gives you something to do afterwards; the casino is always bumping when I've been there.
If you drive to the game, parking in the arena's garage is by far the most convenient route, esp if you don't know your way around. I did this for my first game and once you park, you just follow the mob to the arena, virtually no way you could get lost or mixed up. I can't remember exactly how much it was since it was a few years ago, but it couldn't have been more than $15, well worth the proximity and ease.
You should def swing by Hockeytown cafe at some point too (extra points if its right before a game). The atmosphere is awesome- everyone's in their jerseys, cool memorabilia, good food. Like they said, it runs a free 5 min shuttle to the Joe as well.