Can someone tell me what do I have to do to raise my pet intelligence level?

You can read books to your pet, you can purchase books from the main shops, like Book Nook, or also user shops

There are also stat boosters which add intelligence, but if your pet has lower intel stat, you are better off reading books first as the boosters are expensive Read more about boosters in the battle forums here

if your pet has low intelligence the best option is to play Mystic Spindles. You earn points to buy int, but the points needed go up sharply as the pet gains int, so its only worth it when the pet is very low int.

Books! Of course there are some other ways as some have mentioned, but it's really quite cheap to boost that intelligence with books :)