I was wondering if anyone can tell me the different ways to add more pets to my pet family?
[Tot=thonnya]
The easiest way is to get a gold account. It automatically bumps your pet slot count up to ten. If you abandon/adopt out a pet once the gold account expires, however, you must wait until you renew it to fill the slot again.
After you have maxed out those slots, you can purchase more slots using CSC from the Pets page for (if memory serves right) 2000csc. You can add a maximum of fifteen extra slots through this method.
The third way to gain slots comes from choosing the pet slot totem from the annual Loyalty Box that you earn on the anniversary of your sign-up date each year following your second anniversary.

Just out of curiosity - how on earth do you feed and train so many pets, and keep up with their treasure boxes too? (I hope this doesnt sound rude, because I am honestly just curious)!
I don't actually train my pets, as the battling system has never really been something I pay attention to. Whenever I need to feed them I drop them off at the resort, which I admittedly don't do very often because I don't battle. As for their treasure chests that one is fairly easy. Writing and character design are my two biggest hobbies, so I fill their TCs with items relevant to their stories as I find them. ^^

As far as I can tell, you don't need to feed/train every pet at all. You can put your battle pet in the resort and let the rest starve in boredom with no consequences. :)