LOL, ^^Chick is much braver than I. I can't even imagine the horsehair in the....um...unmentionables
Whenever I'm going swimming, I normally just wear swimming attire myself but my horses don't wear any working tack because all my tack is leather. They get ridden in a halter with rope reins.
The first time I took Dobe swimming, I did have a cheapy bareback pad (no stirrups) that I put on him just to have a bit of traction. Once most of the excess water is off them, they aren't so slick, but going along in the water, it will get between your legs and the horse and it's about like trying to ride a greased whale LOL.
As for the first time trying to swim with them, I stick with the relative shallows (as deep as they can go without losing footing) for quite a while until they are comfortable, then I will start taking them out on very short deeper excursions where they have to swim for a few seconds.
Keep in mind that horses have to work really hard to keep their head above water if you sit on them square as if you were riding on dry ground. If you are going to take them deep for more than a few seconds or if you plan to try to cross a deeper body of water (pond/river), then you'll need to get off and float beside them. Grab a handful of mane and let them drag you along beside them as they swim.