How do I level up my pets? And increase their strength, defense, health and speed?

By training in Riverside, using stat boosters (there's a list in the Battle Forum), or using experience points, that you get from Battling or Battle Quests.

By training them at the training center at Riverside Valley.

You can increase their intelligence by reading to them and also get stat changes via the Zapper V2 on Atebus, but training is easier and cheaper than zapping. c:

Thank you!