On verpets, to find a new pet you must select a location to explore and hunt for one to capture!
Certain pets and colours are harder to find than others and some species do not dwell within this map. However, using certain items in your inventory as 'bait' can apparently tempt the rarer pets of Verpets to approach you, giving you the chance to capture and name them.
Watch out though! Some of the rarer species of Verpets won't give in to being captured without a fight! In these cases you can pick one of your pets to fight against them in a Coliseum 'Quick Battle' - each of you makes a move and the strongest wins.
If it's your first time creating a pet - don't worry, you won't have to battle to capture your pet as you don't have one to battle with at the moment!