The dog ate from it's food dish and drank from it's water dish. - wrong
The dog ate from it is food dish and drank from it is water dish. - wrong
The dog ate from its food dish and drank from its water dish. - correct
As you can see it's = it is, and it is makes absolutely no sense here.
Whenever you could write his or her, you have to write its as well.
(If you don't know its gender.)
Which one is right?
I think. Both are correct but when I talking to use, I usally use "it's". Because it's easier.