Try Guild Wars, it has no monthly fees, we have a very active division here in TOG full of friendly people who will give you plenty of help. There is a thread over in the GW section of the forum for requesting trial keys so you can try before you buy.
If you have Half Life 2 there is a mod called Synergy which adds co-op play, some of the design is a bit patchy but it's still a fun game. There is a similar mod for Doom 3 too.
There are a few Squad based shooters that have co-op too, take a look at Swat 4 and The Rainbow 6 and Ghost Recon series.
Gears of War has a 2 player co-op mode, you can play through the whole game together.
One to keep an eye on is Left 4 Dead from Valve, it's not out yet but it's looking promising, it's a zombie shooting co-op FPS.
There is a huge list of co-op games on Wikipedia here-
Category:Cooperative video games - Wikipedia, the free encyclopedia
That should give you plenty to go on