To obtain the Zelda’s Golden Horse side adventure in The Legend of Zelda: Tears of the Kingdom, speak with Penn at Snowfield Stable during the Potential Princess Sightings! side adventure.
You’ll then need to head north to look for Princess Zelda’s golden horse. As you head north, avoid the 3 headed dragon unless you’re looking for a serious challenge! Further north from the dragon in the North Tabantha Snowfield, you’ll spot the golden horse with 3 wild horses.
Use all your sneakiness (stealth up armor if you have it and elixirs) to sneak up behind the golden horse undetected then quickly mount and soothe it. Once you’ve caught the horse, guide it back to Snowfield Stable and speak with Harlow to complete the quest. As a reward, you’ll get to keep the horse. You’ll also receive a Royal Bridle, Royal Saddle and Energizing Elixir.