How do I increase my pet's stats such as strength and intelligence?
You can increase most stats by going to the Training Center in Riverside. Intelligence can be raised by reading books to your pet or playing Mystic Spindles.
On top of that, you can get extra boosts from random events or playing Scratchcards.
Intelligence is gained by reading books to your pets.
Strength (and other stats) are gained either by random event, or at the Training Center.
There's a Healer in the Sacred Lands who will do it for you, or you can buy Healing Potions.
You can go to the Healer, and she'll heal up your pets for you. :)
Annnnd clearly is much quicker at responses than me, haha!