If you have a waist size <40" you'll be able to get breeches from Dover or Smartpak, or your local-local shop (your instructor will be able to tell you where that is). If you're between a 32" and a 38" waist you'll have a somewhat reduced selection of breech styles, but you should be able to find it. They're all sold by waist measurement. If you're <34" you will be able to find things on the shop floor to try on, and that would be the best thing to do, because all of the styles fit a bit differently. Once you find something that works, you can just order it online (even from Amazon, sometimes!). I like Devon-Aire Versailles breeches the best, for myself, because they're very stretchy and they fit my figure reasonably well. Fuller Fillies is the brand I know of for your more generously padded women. Can get that online, and sometimes the tack shops will carry it too.
I just checked out your barn, and it does look like a show barn. But they've got a heated indoor ring! I am JEALOUS. What that means for you is that you won't have to stop taking lessons in the winter, unless you want to, which is very nice because then you don't have to recondition your body to riding every spring.
Anyway, it looks like an English barn, which means you'll need paddock boots, not cowboy boots. Did they talk to you about what footwear to wear out to the barn for your first lesson? People usually want to wear hiking boots but that's not a great idea - the lug sole becomes a safety issue with the stirrup. If you have a relatively smooth-soled boot with a heel, that would be the best. After you decide (I hope!) that you love this, then you'll go get your paddock boots and half-chaps, and that will do you until you decide you want a pair of proper tall boots. And some people never get a pair of tall boots, and just use paddock boots all the time.
At some point, even if you don't want to show, you'll need to pick a particular discipline (hunters, jumpers, equitation, dressage) because while they all use the same basic skill set, the equipment is a little different, and you do different things with it. I do dressage with my horse. He's actually a really high-level jumper (think of the stadium jumping with the brightly colored fences in the ring from the Olympics), but he's old and his legs are too battered to do that any more without hurting him, so we stay flat and work on flexibility with dressage. I think I saw a picture of Contesstant using a dressage saddle at some point, too. It's a popular discipline for people who like finesse over raw power, and who prefer to stay on the ground rather than spending time in the air.