Minecraft wiki entry for Conquering_a_mountain, associated text: Any common block or ladders, to tower up on cliffs too steep to climb easily.
Scaffolding and twisting vines are good alternatives to ladders, though stumbling upon a jungle or warped forest for appropriate supplies may take a while.
Leather boots, to prevent falling inside powder snow.
Enchanting your boots with Feather Falling may help mitigate harm from potentially frequent falling.
A water bucket in case of any danger; necessary for all missions regardless of difficulty!
A sword, or an axe most preferably Iron at minimum, to quickly kill as much danger (e.g. Goats and pillagers) as possible. (An axe is stronger and can chop trees, especially if you can't see through the Grove forest or don't have any wood)
Plenty of nutritious food, such as steak and bread, to regenerate health.