How to summon. Simple.
To summon you need:
•A non-empty pet card (empty one obtainable in merchant shop, or I can give you a free pet card BASIC ONLY)
After you capture or obtain a filled pet card:
•Press Alt+Y
•Open your inventory, and drag the pet card into the window.
After you put your pet card into formation you should be able to see the pet in the window on the bottom-right of the screen. Summon it and be happy of your new pet.
Some people seem to get confused on how to do this, I cant see why but hey!