–          Trekking Boots (Waterproof)
–          Clothing for hiking (light, breathable, quick drying)
–          Clothing for cold and rain (down jacket and rain jacket/pants)
–          Gloves (light or fingerless)
–          Brimmed hat and winter beanie
–          Sunglasses and sunscreen
–          Sleeping Bag (0F / -18C) or (20F / -7C)
–          Air or foam mattress
–          Water bottle (Nalgene or similar)
–          Backpack
–          Duffel Bag

–          Trekking Boots (Waterproof) or trail shoes
–          Mountaineering Boots
–          Gloves (snow/ice climbing)
–          Gloves (light or fingerless for hiking)
–          Clothing for hiking (light, breathable, quick drying)
–          Clothing layers for climbing (see below)
–          Brimmed Hat and winter beanie
–          Sunglasses and sunscreen
–          Gaiters
–          Backpack (65-75 liters)
–          Daypack (25-45 liters)
–          Sleeping Bag (0F / -18C)
–          Sleeping air pad
–          Water bottle (Nalgene or similar)
–          Duffel Bag

It is important to have your layer system dialed in so you can climb fast while not getting overheated. Everyone feels the cold and conditions differently but I will give you the example of my own system.
 
Lower body – 3 layer system:
–          Base layer: Thermal tights
–          Mid layer: Alpaca wool pants or down pants
–          Outer Shell: Waterproof pants (goretex or similar)
 
Upper body – 4 layer system:
–          Base Layer: Under armour cold gear long sleeve shirt
–          Light Mid layer: Synthetic down hoodie
–          Heavy Mid Layer: 700+ fill Down jacket
– Outer Shell: Waterproof jacket with hood that can go over helmet (goretex or similar)

The most important pieces of clothing/gear for mountaineering are where your body makes contact with the mountain: gloves and boots. I would recommend single boots like the Scarpa Mont Blanc or La Sportiva Nepal Evo or similar boots for peaks that are 5,000 meters up to a peak like Tocllaraju (6,034m) that just goes over 6,000m. For anything higher like Ausangate (6,384m) or Huascaran (6,770m) double boots are best for most people but not absolutely necessary. I personally use the Scarpa Phantom 6,000, because Scarpa boots have a wider toe box for my wide feet than La Sportiva boots. But there are good La Sportiva models also like the Spantik or the G2 SM. As with all mountaineering boots, buy them a size larger so you have room for an extra sock or wiggle room to keep the circulation going.

My all-time favorite gloves for peaks in the Andes are the Outdoor Research Alti Gloves! They have a thick liner for approaches when you can keep the outer shell in your pack, then when the temperature drops or you have to put your hands on the wall, put on the shell. Black Diamond and Marmot also have good options for ice climbing gloves. It is always good to have a lighter glove for hiking or coming down off the mountain when your main gloves may be wet. Another great trekking item is alpaca wool fingerless gloves for hiking or doing things around camp, but I usually leave these at the last camp before going for the summit.

While trekking I use a fully brimmed, light and breathable hat to keep the sun off my face and neck. Something very important in my opinion is a neck buff made of light fabric, that keeps the wind off your neck while trekking. While climbing you can pull it up over your mouth and nose to warm the icy air before breathing it in, it goes a long way to prevent sun/wind burn as well. A winter beanie hat is necessary up on the mountain. Sunglasses are a good idea while hiking and a must up on the mountain where the snow will be reflecting the sunlight right back in your face.

It is important to get a backpack that will fit all of your stuff! Several times I have seen people who have a 45 liter pack, expecting to fit everything they need for 2-3 days on the mountain in it and it usually does not work. I prefer to have a 65-70 liter pack that fits all, has axe loops and crampon storage on the outside. This way you can port everything you need to a higher camp and then empty it, except for what you need on the climb day, the pack itself should only weigh a pound or two. I use the Black Diamond Mission 75 pack and it works great.
If you are only trekking with horse support or do not have to port anything to a higher camp, a smaller day pack up to 45 liters is sufficient.

Any waterproof boots, Goretex or similar are fine for hiking in the Andes, just be conscious of weight, as always with altitude every gram will cost you. My favorite brand of trekking boots is Asolo, especially the models with a rand that goes all the way around the sides of the boot. This eliminates the stitching that can pop if you hit the corners of your feet on rocks while hiking.

A 20F (-7C) to 0F (-18C)  rated sleeping bag is fine for hiking in Peru.
 For climbing in Peru a sleeping bag of 0F (-18C) is perfect for sleeping up to 6,000 meters. It doesn’t get as cold in the tropical latitudes of as Alaska, or the Himalayas. I have even used lighter sleeping bag to save weight and space in my backpack. You can always sleep with an extra layer of clothes on.

These items are generally the same from brand to brand, but just keep in mind ease of use, adjustment, and weight which will all add up to being efficient on the mountain.

In my opinion traditional straight traverse axes should only be used on routes graded F or PD, but beyond that you should use 2 technical ice tools for routes AD and above. Moderate technical tools like the Black Diamond Viper or the Petzl Quark are good for anything but the most extreme vertical ice and mixed climbing. I really like the Black Diamond spinner leash or similar bands leashes of this style to keep you from dropping your axes off the mountain or in a crevasse, they have saved me from losing tools many times.
Crampons come in many different models, but for general mountaineering it is best to have 10 or 12 point crampons. The most important feature you should look for is how easily and quickly you can put them on securely and take them off.