Learn how to distinguish essential mountaineering equipment from unnecessary extras.
Tent and Sleeping Bag
Your tent should be waterproof and big enough to accommodate your group, with a little extra room. Look for the warmest sleeping bag you can find. It should at least be able to handle temperatures at or below -29C. If your sleeping bag doesn't have a hood, make sure you bring a warm hat to sleep in. Never sleep with your head inside your sleeping bag.
While a tarp will keep water from accumulating under your sleeping bag, a foam pad is better to keep you warm. Bring a couple to sleep on at night.
Shovels are important to help you dig out your campsite, move snow and build snow shelters. Ideally your group will have several heavyweight (preferably aluminum) shovels.
Rope is used to fix a line as well as for rescues. Make sure you have water repellant, 45 meter long, 9 millimeter wide rope for two people. If you have three people you'll need at least 50 meters. Bring an extra rope for rescuing.
An ice axe is vital climbers' equipment. Each member of your group should have their own ice axe, plus an extra ice axe for the group. Ice axes are used for ascending and descending as well as self-rescue, which is why it's important to have enough.
Bring ice screws, which can help mountain climbers with rescues. These are screwed directly into the ice and have come along way from the original ice screws, which were pounded into the ice with hammers.
Layer your clothing to avoid overheating or carrying excess clothing. Start with long johns or any other clothing which will wick water away from your body. Next you'll want an insulation layer to keep you warm (not cotton). Look for shirts, sweaters and pants made of wool or fleece. Your shell (jacket and wind pants) should keep rain and snow off of your body.
Gloves should be thin but insulated. You may want to have an extra pair of thick and warm gloves for belaying and rappelling, since your hands will likely be cold. A face mask is also a good idea, as it will keep wind off of your face and keep your face warm. If you'll be mountain climbing in very cold conditions you'll need an "expedition" parka and pants. These will keep you warm in the coldest of conditions.
Make sure you have extras of most of your clothes (though not the expedition parka and pants). You'll need backups in case your primary clothes get wet or you end up on the trail longer than you expected.
In addition to the boots you'll wear, make sure you have thick wool or synthetic socks (not cotton) and crampons to provide traction on uneven terrain.
Choosing the right mountain climbing equipment is a challenge. You need a lot of stuff but don't have a lot of room in your pack. Learn to stick to the essentials.