You can get to your pet's yard by going to the Neighborhood Page and choosing the "Build" icon, which looks like a hammer and nail. The yard is locked when you first get it.
Once inside the Build Mode, you can decorate both your house exterior and your yard. You can add doors, windows, roofs and other items to improve the outside appearance of your house. These items are somewhat expensive to be purchased and can only be placed on the square house.
Unlocking and Decorating your YardSquirrel Tree Home. Once you have this item your yard is freed up to place items into it.
You can move the location of your house on the yard by clicking on the house and dragging. Items can only be placed within the light green squares. You cannot put items on top of other items.
Items that you buy for decorating can be sold and gifted, but so far cannot be recycled. Some items for your house are collaborative items, but for some reason do not show up in the Collaborative toolbox when you are inside your home. See the collaborative page for a listing of these items. You can request friend's help for these items only when you have unlocked the yard and are in build mode.
Items that can be placed on your yard cannot be placed on the house exterior.
Yard Expansion Levels
You can pay 10,000 coins per level to extend the size of your yard. Currently, the highest level of expansion is Level 6, so to fully expand your yard it would cost 60,000 coins. It looks like further size expansions will be forthcoming.