There are several books out there on Massage for Cats, and Massage for Dogs, etc - I nearly published one, but I decided that YOU know what kind of touch your pet likes. My dog loves his tummy being stroked (effleurage), as do most dogs, and my cat, her shoulders being rubbed (bit of friction) - until she swipes me one!
Just observing animals at birth shows us how massage helps the little ones - not just to remove the meconium and the foetal sac, but to stimulate the passing of faeces and urine when they are little - in the first few weeks after birth. The mother dog/cat will lick (stroke) their babies little bottoms and willies to make this happen. And they clean them all the time by licking them, don't they? That's massage.
Birds also like their feather's being stroked - some of them anyway. My parrot loved his tummy being tickled, until I found out that that is where the genital organs of a bird are and no wonder he considered me his mate - and bit anyone who came near me!
So, there is lots to learn about massage, and what better way to start than with your own baby? What a start in life you can give them by learning infant massage!