Hm, difficult question. I never had the need to master my XBox Controller. It always felt more natural. Well, first you have to keep in mind that not every game is meant to be played with a controller even if it has that option although it got better. With this said you should probably start with games which were meant to be played with a controller. In general this counts for most platformers, 2d beat them up and everything ported from console (makes sense). Some puzzle games work with controller too and even some untypcial genres (examples would be Wasteland 2 (RPG), Outlast (FP), Borderlands (FPS), i think Metro 2033 too). I’m not sure if you want something easier or also with more precision/faster movement in order to train the controls. But I will take a quick look through your library (checked SG wins and unplayed, can also check the others if you wish):

• Phantom Breaker: Battle Grounds: Definetely easier with controller and most of the time the controls are also on the easy side. Not much precision needed or being overly quick
• Half-Minute Hero: More fun with controller and can become quite hectic but it is also on the easier side control-wise
• Miracle Fly: That one starts fairly easy but can become really hard quite fast especially in the precision part. Also pay attention that you use the thrust button more intuitive for you. I can’t even remember if I played this one with controller but I should have done it.
• Shadow of Mordor: Never played it myseld. But it should be playable quite good with controller. And since it is AAA at least a part of the game should be accustomed to the casual gamer.

• Never Alone: Quite easy platformer and most time slower paced.

• Resident Evil 5/Revelations 1&2: Also meant for controller. Aiming is slightly more difficult but the rest of the movement is more handy with controller. But has lots of tense moments (if you are not accustomed to the games) and might stress you out.
• Toki Tori: Puzzle Platformer and you have lots of time most time. Minor precision needed
• Two Brothers: Didn’t play it but it was designed with controller in mind and may especially helpful since you control both brothers with each one of the analogue sticks (i think?)
• Ys Origin/Oath in Felghana/II: Also meant for controller and really fun games but they are for the most part brutally hard (especially the bosses, high timing and precision needed). You will either gain incredible control or rage problems.
• Amnesia: The Dark Descent: You have to check but it should be playable with controller and since you can’t shoot it should be easy most time.
• Braid: Puzzle Platformer, didn’t play it so I can’t argue for it. But controls should be good for controller
• Fallout New Vegas: Works also really well with a controller (coming from consoles). With the VATS it is really easy to fight and the normal movements makes no problems as well.
• Hyper Dimension Neptunia: Typical JRPG. Easy controls

There is probably more but I just didn’t play enough games to know the controls of them -_-

My rule of thumb is if it is a FPS game then Keyboard/Mouse all the way.
Platformers, rpg, racing, controller tops it.
Above is a good list of games that should be used with a controller.

I just got an Xbox controller and before years ago I only used PS3 controller.
I hate the location of X on the Xbox controller, every time a “press X” message appears, always will press X on the controller as if it was a PS3 controller.
So you are not alone who has issues.