Q: Computer won't recognize Firewire/Thunderbolt connection

I just purchased a new iMac and am trying to move my stuff from my old Mac to the new one.  I bought a Firewire 800 cord with the Thunderbolt adapter, but my computer won't recognize that it's plugged in - in fact, neither computer recognizes the connection.   I've also tried booting the old mac in target disk mode, but my new computer won't mount the old drive the way it's supposed to - I'm assuming because it doesn't recognize the connection.  No matter how I try to connect the two computers, one doesn't seem to "see" the other aside from Wifi, which I don't want to use because I have so many pictures and videos.  I'm mainly concerned at this point about how to efficiently move my iPhoto library over, but without a fast connection, that will take days.   Does anyone have any suggestions?  Am I doing this wrong?

Jennifer