After you defeat an enemy at Hyrule Castle, you set out to visit the rifts in the lands connected to the Goddesses of Hyrule.
Prepare to Visit the Three Lands
Before you visit the three lands, if you have already visited Hyrule Ranch and completed the quest Runaway Horse, you should go to Hyrule Castle and talk to Impa to start the quest Impa's Gift.
If you haven't already visited Hyrule Ranch, it is recommended to do so. Start by going to the southern gate of Hyrule Castle Town, and go clockwise around the castle wall until you reach the bridge to the west. Cross it, then go south along the dirt path, then when the path splits, go west. When the path splits again, go north and follow the path to the west. When you reach a sign in the road, first go west and check on the Waypoint to add it to your map, then go east to the road sign again, and go south. You'll arrive at Hyrule Ranch.
Hyrule Ranch
In the building at Hyrule Ranch, you can buy Fresh Milk. Then, if you haven't already, talk to the man west of that building to start the quest Runaway Horse.
After you complete that quest, go back to Hyrule Castle and talk to Impa to start the quest Impa's Gift.
Visit the Three Lands
The three biggest rifts are in the three lands connected to the Goddesses of Hyrule: Eldin Volcano, Faron Wetlands, Holy Mount Lanayru.