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Even though the most important things are fitness, good health and good sense, after looking at picrel I want to open with some elements of the correct shoes which should be an easy point to get the thread going.
For mountainous terrain shoes are a matter of safety, inadequate shoes will result in high potential of injury or even death:
1. They must cover the ankle and have a degree of rigidity both in the upper and the sole, so to offer the support to the ligament, this helps in injury prevention
2. They must have a deep aggressive thread so to grip the ground safely, picrel is some examples. A fall might mean death and worn shoes have been responsible for lethal falls on grass and in good weather. Even this won't be enough on frozen ground or snow.
3. They must protect from elements effectively, that is have a good degree of water resistance.
FIT:
Buy boots in person. Try the boots on with the socks you will be using outdoors. Push your foot towards the very tip, a finger must be able to fit behind the heel. There should always be some wiggle room for toes. Once laced up, the foot must not move inside the shoe, especially the heel must remain firmly in place, shoes too roomy cause painful blisters.
Try walking around a bit, if the boots is uncomfortable or has pressure points, try another model or another size.
Socks depends on the style of boot chosen, both depend on the activity and environment they will be used in. Generally a medium weight knit wool sock is the standard, wearing a thinner sock underneath will prevent blisters and preserve the sock too.
There is much more to say, these are the most basic elements in boot choice and will suffice for now.