"Me too" is perfectly grammatical, and you shouldn't pay attention to so-called "grammatical fanatics". It's similar to answering the question "who is it?" with "it's me." You wouldn't say, "it's I" unless you wanted to continue with "(it's I), the King of Monarch-land" or something like that.

"I, too" is not correct. "I, also" is also incorrect. For formal situations, "me too" might be a little casual. The following are more formal:

I do as well
I do too

(some people might say "me, also" is correct. It's marginal for me)