After you emerge from the Mining Cave and explore Great Sky Island further, you discover Pit Cave.
Explore the Cave
Near the entrance of Pit Cave, you will find Spicy Peppers. Be sure to gather these. In the cave, talk to the Construct to learn about making food that keeps you warm. You can cook the Spicy Peppers in the cookpot, either by themselves or with other ingredients, to make food that will give you temporary Cold Resistance when you eat them. These peppers don't give Cold Resistance until they are cooked, so be sure to make food with all of them.
Go up the slope in the cave and some Keese will attack. Eventually the path will split. Go to the right to find a Bubbulfrog. Defeat it to get a Bubbul Gem.
Collect the Brightbloom Seeds, hit the ore, and continue onward to find a stone wall that you can break with a Rock Hammer. This takes you back to the entrance of the cave.
Reach the Upper Exit
Now go south through the cave, but this time go left at the fork in the path. Climb up the walls and you will reach a cave exit up above to the cold area. Eat one of the foods that you cooked with Spicy Peppers (use the cookpot at the entrance of the cave if needed), then go out through this opening into the cold area.
Reach the Next Cave
You can optionally go west to find a structure of cubes. Pick up the smaller cube segment and rotate it to complete the large cube. You will find a Korok.
Go east and a White Chuchu will attack. Kill it from a distance if you can, because if you are close to it when it dies, it will temporarily freeze you and you have to wiggle the control stick to escape.
Farther east, you can talk to Rauru. Then go north and defeat the Constructs. If you have flint, you can drop it next to the cookpot and hit it with a metal weapon to start the fire.
Go north and down the slope, defeat the Chuchus and Soldier Construct and take the Flame-Emitter Club that the Construct was weilding.
Then enter the cave. This is Bottomless Cave.