Private Rooms will unlock at level 13, after completing a quest for Trina.
There are 3 different rooms:
You can view your Friends' Rooms by tapping on the Friend icon in the lower right corner.
Each room has different furniture that can be upgraded. To see what can be upgraded toggle the switch at the bottom of the screen. All the items that can be upgraded will now have an upgrade icon. Once you unlock the ability to upgrade the item you should be able to choose between the different available upgrades.