If a pets stats are below 150, they can be trained all the way to 150 on a training ele. Raising it above 150, or if it's already higher than 150, requires leveling the pet. I can't confirm if it's every level gives a chance for a stat increase, or if it's every 5 levels, though. Once the pet is level 40 points can be added to str/dex/int also, but if you're leveling a pet for like pet wars they won't ever make it past 30.
Also, as a side note. Pets that are tamed with equal Dex/Stam will have those two stats tied to one another. For example if you tame a animal and it has 30 dex/30 stam. Every point of dex gained will also add 1 stamina. Now, if the dex/stam is not equal when they are tamed, or if you add an ability point into stamina, it won't be tied together anymore, so it's generally better to wait to add stamina until the pet is max level if they are tied together.