It's super important to make sure you've got all the gear you need for a mountaineering trip. Some little things are easy to overlook, and that could leave you totally screwed. Bad screwed. I'm sure that will never happen to anyone ever, especially not me. Or you. Anyway, about mountaineering gear...
You're totally going to want a four-season tent, a sleeping bag (most people prefer down for it's packability and warmth) that's rated to at least -30 to 20 degrees. Slapping a inflatable or closed-cell foam sleeping pad under that sleeping bag will totally help increase your thermal efficiency, too. Of course, you'll want a white gas stove, matches or a lighter, fuel, and a cookset with eating utensils: you're gonna get hungry. I promise. I think you should bring a knife or a multi-tool everywhere, so bring one of those, to.
You're going to want a pair of mountaineering boots, either leather or double plastic, and a pair of 12 point crampons and some waterproof gaiters to wear with them. You'll look like a fabulous robot. A pair of trekking poles never hurt anyone, and are great for picking up trash on the side of the highway when you're not on the mountain. And, of course you'll want all your climbing ropes, some daisy chains, runners, a harness, a helmet, and protection, including nuts and screws. Avalanches are nothing to mess around with: bring a probe, a shovel, and an avalanche beacon.
All the stuff you wear on your body will make the world of difference. Waterproof, lightweight, and breathable outerwear is key. That means a hard shell jacket and pants. Maybe Gore-Tex. So fancy. You'll definitely want an insulation layer, so an insulated synthetic or down jacket is a must. Pack a fleece or a wool sweater while you're at it. I bet I sound like your mom. Make sure you've got your base layers, too. I bet your mom calls them long johns.