If you want to start a new game, such as to experience the magic of Calico from the beginning with the Pawsome Update, you need to delete your save file, which is going to be hidden away as follows for Windows 10:
C:\Users\USERNAME\AppData\LocalLow\Peachy Keen Games
Once you're in the folder for your windows username, you'll need to select "View" from the menu on the top and put a check on hidden items.
Once you see the Calico folder in the AppData area, you can just delete the whole folder as the game will make a new one once you launch it again.