It's no secret that climbing is one of the best ways to get around The Legend of Zelda: Tears of the Kingdom, but with Stamina limitations, it can be hard to scale tall structures before Link becomes exhausted. Climbing Armor, while not a complete solution to this problem, does help Link climb faster, which can help stretch out his max Stamina.
Unfortunately, the Climbing Armor is separated into three pieces, and all three will need to be found and worn to get the maximum benefits of the outfit, but each piece does have a small boost, so even one piece is better than nothing. The full Climbing Armor set includes Climbing Gear, Climbing Boots, and a Climber's Bandana, all in different locations.
Climbing Gear Location
The main piece of the armor set is the Climbing Gear chest piece, and it will be found in the North Hyrule Plain Cave, which is Northwest of Lookout Landing on Zelda: TOTK's massive map. Link should immediately encounter two Like Likes upon entering, indicating he is in the right place. Once the enemies are defeated, Ascend through the platform the further one was on and keep heading up until reaching two waterfalls. Head up the right side of the waterfalls until reaching the upper waterfall. Instead of heading to the top of it, head through the upper waterfall into a small room where the chest is.
Climbing Boots Location
The Climbing Boots will also be found in a cave, but this one will be near the Zora's Domain and is called Upland Zorana Byroad. The first area will split into two separate paths; take the right one. Next, head through the gap in the wall to reach another path where Link will cut through some hanging vines before reaching another open area. There will be some water to cross, which Link can do by any means available to him. Once across, open the door using TOTK's Ultrahand ability to drain the room of water. Once it is all gone, drop the door and investigate the room to the right to discover the chest.
Climber's Bandana Location
The Climber's Bandana, to no surprise, is also in a cave. Specifically inside the Ploymus Mountain Cave, which is also near Zora's Domain in Zelda: TOTK, but to the East this time. Fitting to its reward, this cave will require a lot of climbing for Link as he has to navigate across various platforms to reach the end of it. So first, equip the Climbing Gear already acquired to boost the climbing speed and then alternate between the left and right side platforms. Once at the end, Link will find a room containing a chest with The Legend of Zelda: Tears of the Kingdom Climbing Boots inside. Now the Climbing Gear Armor set is complete.