Your two-room house is where your home base is. Inside your house you can connect to the Nintendo Wifi Network, save your game, and switch companion pets.
If you enter your house and then leave, the day cycle will be the same as when you entered. If you did the same thing in the fishing areas the day will cycle. For example if you entered the Pond in the morning and then left, the day will be afternoon. If you entered your house in the morning the day will still be morning when you leave.
In the northern corner of the house is your Futon. If you stand on the bed and press A you can choose to rest. Resting will restore your hit points and the day cycle will change to morning.
The black telephone in the eastern corner connects your game to the Wifi system, where you can participate in the wireless Fishing Tournaments. Make sure you don't have any fish in your basket before you attempt to connect.
The doorway next to the black telephone leads to Yunna's room. If you talk to Yunna you can save your game. River King DS has 3 save slots you can use. Even if you finish the game and catch the River King, you can still talk to Yunna to save your game.
There is also a trophy shelf inside your house. As you complete tasks in the game you will be rewarded with achievements. The trophies you earn have no affect on the story.