I was reading my pet a book, and when i was done it said she gained an alchemy point...

What do they do?

Alchemy points are used to learn spells at the secluded shrine.

Sweet =3

I'll have to check that out.

How do i get them? Just every time my pet hits a certain intelligence level?

You get them by raising a pet's intelligence. I don't recall the ratio, but for every few points of intelligence that pet gains an alchemy point.