The pet you want will be mostly based upon your build. Stats are more desirable over pet function. That “second slot” is for trophies that you receive from the chests that you win in the arena. The green pets are Eternal pets, and the red are legendary. When an item is eternal, its stats are doubled. When farming, I use an imp with beneficial stats. You will develop your own preference. 
It doesn’t matter whenever you obtain a pet, either. Just farm away for everything.
You’ll grasp everything eventually.
Just ask on the forums (personally or openly) when you have a question.
Refer to this guide made by a cool guy (we won’t mention names(Skaul(me))) for tips:
Welcome to the world of DQ. 
Enjoy your stay, and make sure your floatie functions correctly. 