Me too in Korean is 나도, is you too = 너도 ?
Is this gramatically correct ?
And I rarely hear people say it formally.
So what do they say, in a formal situation ?
Here's some formal version scenarios.
A: 요즘 감기 많이 돌아다니네요. 저도 걸리고 말았어요.
B: 아 김 선생님도요? 저런...
A: 박철수 씨, 나도 이번에 회사 조기 축구회에 가입했네.
B: 과장님도 가입하셨어요? 저도 얼마 전에 가입했는데요.
The tricky part is that Korean doesn't really have a word for "you" in a formal situation.
당신, which might appear to be "you", is never used in a formal and polite context.
So you must know how to use 선생님, 사모님, 님, XX 님, XX 씨, and the titles of people.
Really? we use both words pretty much.
ex) A: I'm so hungry / 나 엄청 배고파
B: Me too / 나도
A: 나 서울 살아 넌? / I'm living Seoul, you too?
B: Yes, me too / 응, 나도
However we normally use 넌? rather than 너는?
Yup, 너도 is "you too", pretty much.
Are you a student, too?
The formal version is 당신도, but Koreans don't use this too often. You're better off using the person's name or the person's title (like 선생님).