In the Legend of Zelda: Tears of the Kingdom, you can receive the A New Champion’s Tunic quest by visiting Link and Zelda’s house.
To reach it, travel to Hateno Village in eastern Hyrule near the Necluda Sea and make your way to the Firly Pond. Cross the bridge there to find to arrive at their house then head around the back. There you will find Zelda’s Secret Well.
Climb down into it then look for her desk.
Examine the book on it to find Zelda’s Diary then choose the read it. An entry will mention that she ordered a new and improved Champion’s Tunic as a present for Link to thank him for his support. It also says that she hid the tunic in the throne room of the castle so it remains hidden until after their investigation. This will officially kick off the quest.
Now that you have the quest, travel to Lookout Landing Skyview Tower in Central Hyrule.
You’ll want to use the tower to launch Link into the air and glide over to the castle. It would help to have a bigger stamina bar or the Gliding outfit to make reaching it a bit easier. After you land, follow the main path to reach the Sanctum. On the right and left side of the room, is a set of stairs leading up to the balcony in front of the throne. Make your way there to find two unlit braziers. Light them both using a Fire Fruit.
This will cause the statue behind the throne to move revealing a secret cubby hole. Climb up on it then look inside to spot a chest. Open it up to receive the Champion’s Leathers. This will complete the quest. While you’re here, you might want to check out the upper level of the Sanctum. There is some good gear up there that could give you an edge!