After several months I finally have a working CD player installed in the van. It turns out I was just missing a lead to tell a remote amp to turn on, as I had suspected.
It wasn't all that easy, though. The adapter kit added so much extra wire that the new radio didn't want to fit in the dash. Trying to work the wires slide back further in to make room I busted a few wires off of one of the connectors, and ended up having to route around the connector. This means I won't be able to simply put the old radio back in one day. Not only that, but I blew a fuse when the connection broke, because at this point I had already hooked everything back up to make sure it worked.
That's all taken care of now, but I'm still not done. This model van doesn't have a nice generic faceplate you can just pick up anywhere, and the radio still doesn't fit well.
Fortunately I have seen kits online for a reasonable price that I should be able to use, but I'm gonna wait to buy one until after I figure out a way to anchor the radio in a little better.
FYI, the post title has a double meaning, as Rejoice by Audio Adrenaline was the first song I heard on the new radio when I finally got everything put together.