Bridle size can really vary between brands. I find that bridles marked "cob" or "large pony" fit smaller, older-type quarter horses well. The bigger framed new-style QH's and Appendixes are a better candidate for a "horse" size. Taking measurements of your horse and comparing them to different bridles may be your best bet until you get a feel for "her" sizes. As far as style goes, I think one or two ear headstalls are pretty popular in the western show world. Most of them don't have throatlatches, however, you may want to look into one that does if you plan on doing much trail riding. Bridles without throatlatches just come off a little to easily for my taste.
As far as the bridle goes, if I were in your shoes I think I would at first get something like this: Harness Leather Trainer Headstall - Smith Brothers
, for daily use, and later get something more showy, like this: Tory Oklahoma Double-Ear Silver Headstall - Smith Brothers
, for showing. Some show saddles, esp WP saddles, will come with a matching headstall.
Reins are really personal preference as far as the thickness of the leather. I personally have smaller hands, so thinner reins give me a better feel than thicker ones. Take a trip up to the tack store and hold a few different thicknesses, and get an idea of what feels good to you.
You'll probably want a saddle. If you are planning on showing WP and reining, you may want to decide which you will probably show more, since they use slightly different styles of western saddles.
WP - Western Show Saddles
Reining - Horse Reining Saddles
A reining saddle might be better if it's also going to be your "daily driver." I feel guilty if I ride in my silvery pleasure saddle outside the show pen.
At very low levels and open shows you could probably rein in a WP saddle or ride WP in a reining saddle, but much higher levels than that, not having the "correct" saddle may cost you points. As far as sizes for saddles, unless you have some experience fitting them, I'd recommend finding a saddle fitter in your area, or a knowedgable buddy to help you fit it to your mare. Full QH bars aren't always necessary, but often are. I have a semi-QH tree pleasure saddle that fits some QH's but not others, and I've even seen a couple that needed a narrower tree than the semi. Someone at your local tack store can probably assist you, as well.
That was probably more info than you ever wanted, ha ha. Hope that was helpful to you. Good luck on buying your horse! Can't wait to see pics!