You're gliding down a snow-covered mountain at up to 70 mph on a snowboard or up to 100 mph on skis. Wax makes that possible. So, make sure your board or skis are waxed. All mountains rent gear. But if you're going to buy, here's what to look for: Length and width. Snowboards are measured in centimeters, 155, 160, 165, etc. The longer, the faster. The shorter, the more mobility. Figuring out your ideal length takes time. Boards are also bent inwards in the center (camber) or bent outwards (convex). Camber allows for smoother movement. Convex allows for sharper turns. And of course, there are blends of both. These designs are important depending on the type of snow you're riding on. East coast snow is hard and crunchy. Midwest and Westcoast snow is soft and fluffy. All of this stuff is extremely important if you intend to ride a lot. Skis are essentially the same, except for each foot.
You can wear your regular sneaks to get you to the lodge and your transpo. You're not going to keep them on long anyway since you'll be switching to your snowboard/ski boots.
All mountains rent the appropriate boots. But if you're going to buy, here's what to look for: Snowboard boots are soft and can feel like extra rigid sneakers. Good fit means secure, but not too tight. And that depends on the way you secure them (there are different mechanisms to secure a snowboard boot fit. It's a matter of what you prefer, so try them all out). To secure your boot to a snowboard, you need binders. Binders are as varied as the boots and you wouldn't worry about these unless you're a professional snowboarder.
Over the knee wool socks. They will always get wet, regardless of how new or waterproof your boots are
Face protection (i.e. baklava)
Eye protection (i.e over the eye visor)
Headwear, like a beanie cap.
Warm (and waterproof) gloves or mittens
Hand warmers recommended
Although you'd look ridiculous, if you're just starting out, knee pads are lifesavers. Anyone staring awkwardly at you is just upset they didn't think of it.
Oh, and a backpack to stow your favorite stimulation on or off the mountain. We don't encourage getting stimulated while engaging in dangerous activities. But we do want you to get your backpack from GORUCK. Disclosure, we're an official affiliate and get a commission when you buy from us.