What specifically are you having difficulty overcoming with the PC controls? Is it the mouse, keyboard, or both?
If you are having trouble with mouse, I assume that it's because it seems like a different way of aiming. Try setting the Sensitivity in the options of the game up or down. Try to find what feels right for you. Also, make sure you have a decent mouse. A ball rolling mouse just ain't going to cut it. Get a laser mouse with at least a scroller. Side buttons are also a plus.
If the keyboard is the problem, I would highly suggest that you use the WASD method of controlling. Instead of the arrow keys to control your movement, use W for foward, A for left, S for down, and D for right. Shift, which you use with your pinky, is mostly used for crouch. R is for reloading. E is generally for use. The number keys are usually for your weapon selects. Most FPS on PCs nowadays just default to WASD controls though.
I know that it can be difficult to transition from a controller to a mouse. However, it just takes time and practice. The bonus is that once you get good with it, it will provide you with much more accurate shooting than a standard controller can provide.
Good games to get into...really depend on how good your PC is. If your PC is good enough, I would recommend the latest titles.
Half-Life 2 is always a fantastic title that can run on a wide variety of PCs.
Counter-Strike: Source is a good online game for competition. It's not quite as difficult as the original Counter-Strike, so you won't do as badly when you first start playing.
Team Fortress 2 is also a really fun online shooter. It doesn't take much skill, so it might be good for you as you start to learn PC controls and try to get better with them.
Battlefield 2 is also another solid online shooter.
Farcry is a pretty decent single-player.
Call of Duty (the first one) is great single-player that should run on most modern computers.
Deus Ex is still one of the best FPS of all time and can be a really great time so long as the old graphics don't bother you.