Official game message boards
One of the best places to find other people to game with is on the game's official message boards which are maintained by the developer. On these forums, you can register and post in "add me" threads where other players will find them. If you go this route, remember to include a link to your Facebook profile otherwise they will not be able to add you as a friend. You also should ask them to put in the friend request what game(s) they play, so that you know who they are when they add you as a friend. You should be able to look through the threads and find friends to add as well.
On the wall of the game's fan page
If you go to the fan page for the game that you wish to find friends for, you can comment on the status messages, wall, and review page with "add me". You will see many other people doing this as well, so you can feel free to add them as friends. This is probably the fastest way to find friends. Sometimes, simply clicking "like" on a game's wall post is enough to cause a barrage of friend requests to fly in from all angles.
In "Add Me" groups
Most games (at least the popular ones) on Facebook have various "Add Me" groups that you can join to find other people to play that game with. You will have to join the group, but once you have joined you have unlocked a ton of people who play the game game as you. It's a nice way to exchange tips and tricks on the games too. See FarmVilleAddMe for an example of this.
Another perfect way to find friends to game with is by commenting on this post! Leave a comment with your Facebook profile link and tell others what games you play. There's a good chance you'll find plenty of friends this way!
General Tips and Warnings:
- Remember that if you post on forums and external sites, you will have to delete your posts in the future if you want people to stop adding you as a friend.
- When you're adding strangers, it's best to make a special Facebook list/group for them and do not share any personal information or non-gaming status messages with them.
- It is possible that spammers will add you, so use careful consideration when accepting friend requests.