Your home is really basic when you start and it will stay like this for the majority of the game. It is different depending on the type of Pokémon you have as your main Pokémon
After you have defeated Articuno, you will get confronted by a Wobbuffet & a Wynaut who ask you to go to a dungeon to get a trio of Mankey.
Defeat & Bring back the Mankey and your home will change to a standard construction home. These are all identical but have a small face
This however is a construction stage and the Mankey will not start work on it until they get their berries back. They are on the 9th floor of the dungeon they were in and looks like a cookie. You will have to go through it a couple of times to get the amount they want however.
Once you have returned the Mankey's food, they will complete your home and it will look just like your main Pokémon such as mine being a Bulbasaur

 
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