Ice climbing requires a surprising amount of equipment. First off there's all the gear to actually climb the ice, then there is the gear to protect the climb, and finally there is all the equipment you wear to stay warm in temperatures potentially below -20C. This article details the equipment we found to be most useful while ice climbing in Canada one January. We were generally climbing 3+ pitch routes, but we also spent some time both cragging and doing longer routes. Conditions varied from -25C (In Maligne Canyon) to 0C (On the Weeping Wall).
Axes - you will want two technical axes. Some popular options are the Petzl Nomic (see here for a review) or Quark, Black Diamond Viper or Cobra, or the Grival Quantum Tech. Generally, these are used with umbilical cords or entirely leashless.
Crampons - a set of technical crampons are a must. These can be either dual or mono point. I think it is fair to say dual points are more popular and versatile on ice. Once again some popular options include the Petzl Linx or Dart, the Grivel G14 or G22, and the Black Diamond Cyborgs.
Pack - we used 30-40l packs. When doing a multi-pitch route, the leader either carried the lightest pack or none at all depending on the difficutly and length of the climb. When climbing, the pack basically just contained belay jackets, food, drink and abseil tat. Each climber generally carried their own assortment of gloves (belaying, leading and seconding) on their person to stop them from freezing.
Personal - harness, daisy, auto-locking belay plate, prussic (attached to pack, while climbing so it can be easily hung off anchors), and a helmet.
Rack - 6 x draws (possibly all screamers depending on your style), 4 x 60cm extenders, 12 x screws (potentially 4 less if bolted anchors) and a v-threader. We usually carried 2x 10cm screws, 5x 13cm screws, 4x 15cm screws, 1x 21cm screws. The 21cm screw for v-threads.
Anchor Kit - 2x equalising set-up: A 120cm sling + two locking carabiners. In good ice we used only two ice screws at each anchor.
Drink - 0.5 to 1.5l of thermos capacity each depending on temperature and climbing style (less for a multi-pitch route, more for a crag near car). Remember to rehydrate each night and morning as you can get away without drinking enough for days but it eventually takes it's toll.
Food - this comes down to personal preference, really. We found a sandwiches combined with some bars and potentially a gel worked well. We also carried chocolate.
See here for a detailed article on layering while ice climbing in Canada.
Gloves - leading, seconding and belaying.
Leading gloves are the lightest weight you can get away with one the day - dexterity is key as it will speed up your placements. Less bulk on the palm also leads to less over gripping and thus better climbing. I generally used Sealskinz (see here for a review).
The seconding gloves are probably insulated and should keep your hands warm and dry while seconding, but still allow you to clean gear. It’s worth looking around and paying for a good set as these will become your leading gloves in really cold conditions. I used Black Diamond Punishers and was very happy with them.
The Belay mitts should be insulated. These are just designed to keep you warm while standing around belaying. Get the cheapest ones that will do the job (see here for more details).
Boots - There are basically three options. I’ll list them in order of suitability.
Insulated winter boots with built in gaiters. These are warm, while being lightweight and delicate. These are a great option as your feet wil still be warm enough but you'll be able to achieve more precise foot placements than in a double boot. Popular options include the La Sportiva Batura, the Scarpa Guide, and the Mammut Nordwand TL (see here for a review).
The next best option is a double boot. Although sufficient, these boots are heavier and bulkier than the single insulated boots. These should also be combined with a set of gaiters for powder plugs to the base of less frequented climbs.
Finally, we have the lightweight insulated boots combined with insulated gaiters. This is the student option. These boots will be cold in a Canadian winter and you will need to be keeping an eye on your toes, but you probably already have a pair, and if not they are cheaper than the other two options. I used La Sportiva Trango Extreme's and insulated yeti gaiters. These can be used with automatic crampons.
Bottoms - I would recommend thermals (one piece is ideal), soft shells or hard-shells depending on the wetness of the day. If packing one I would bring hard shells. The layering article goes into more details.
Tops - thermal (once piece is ideal), micro-fleece, light synthetic jacket or soft-shell, light hardshell and a belay puffer. If choosing between the synthetic jacket or soft-shell, I would take the synthetic unless you are planning on doing a lot of mixed climbing.
Hats - a balaclava and merino skull cap is sufficient, or even just a skull cap if combined with a hooded base layer.