I always try to encourage people to try the discipline opposite of them. Your western bridle is fine for lessons, just switch to a snaffle is you ride in anything else. Also, an english saddle is necessary (search consignment first) along with a well fitting girth. Then you need at least one contour pad. The square pads for colour is optional. Depending on the buld of your horse, he/he might need a breastplate or a crupper but I doubt it if you haven't had problems with slipping saddles (if you have, look into them). If your horse carries himself/herself like your icon, a martingale will probably not be necessary. Do you plan on jumping or just getting into the english ways?
Also, no matter what, a riding helmet. Everytime you get on your loyal steed - helmet. And a certified riding helmet at that. I don't care if your head is big, so is mine. I don't care if you don't like the feel, so what. Or any other crap reasons. Helmet!!! (I've heard too many excuses so I'm covering them right off the back)
I hope that helps! Feel free to pm me anytime for help. I am currently helping two good friends transition from western to english and I ride both so I'm not biased