I'm not sure if the tutorial goes into this at all, but this would've helped me early on:

I bought a new house! Now, how do I move in?
To move houses you must first remove everything from your current house.

Pixpets must have returned from hoards and be fully rested in order to be removed.

All decorative items, including wall and floor skins, must be returned to your inventory.

Plants must be returned to inventory, or destroyed and the plant pots returned to inventory.

All pumpkin slots must be empty.

After everything is removed, go to Housing > My Houses, select the vacant house, and click Move into this House.