There are several quests that you will come across while exploring the area and Rito. Many of these will be relevant to the area, as is the case with the side quest, Crossing The Cold Pond. The guide below goes over how to cross the lake, gather ten Chillshroom, and make ice bridges in The Legend of Zelda: Tears of The Kingdom.
How to Cross The Cold Poool
Once you leave the lodge at the bottom of the mountain, you will come across a few mountain trails. On your way to the summit, you will run into Verla, who is looking to gather food for Rito. He tells you that he needs to gather ten Chillshrooms inside the cave ahead but that due to his hurt wing and the freezing temperature of the lake, he can’t proceed. Grab the Ice Fruit to your right and head inside. You will find the first few Chillshroom as you enter the cave.
You will now need to proceed across the lake. There is a raft at the end you need to get to. The key here is to equip your bow and fuse your arrows with Ice Fruit. When these arrows come into contact with the water, they will freeze it, creating an easy path for you to reach the raft.
Once you are across, grab the Chillshrooms and head back across the bridge. Speak with Verla to be awarded a Purple Rupee for your trouble.