I never played games with a controllers, but I just recently got a wireless PC one, so I might give it a shot. However, I'm not sure which games will I be able to play with it! Is there a centralized source of information, or am I supposed to browse gaming forums/ask people to find out?
When a controller is not an option, you can use one of many programs that will map controller buttons to keyboard actions. This will allow you to use your controller for multiple games that might not otherwise support a controller.
Your wireless controller (you did not specific a brand) may come with a program that already allows you to do this. Here is another option I found after a quick search:
... you just need to get past the absolute HORROR of the user interface.
If you use Steam there is a "Controller Enabled" graphic displayed in the right column near the bottom of the game page in the store. Also here you will find out what Steam features it supports like Cloud Saves or Achievements.
If the game you're interested in isn't on Steam, then check to see if there is an Xbox 360 version of the game. In my experience those usually leave the controller option available on the PC, however; its not a guarantee.