It's out! You can now access the contests page from the "town" section of the menu. Best in Show is the only contest available right now, since we don't have enough users to have multiple types yet; and people would technically win by entering empty contests, which shouldn't be. Residence and the like will come later too since, once again, it would need more users than we have now to work properly.
Make sure you enter your best pets! Best in show judges from all the stats combined.
Training needs another day or two to be refined, but remember that you can currently breed for better stats!