“So do I” and “neither do I” are a form of response not easily understood by learners of English and they are usually replaced with a shorter, more common, and more informal answer “me too”.
Use “so do I” or “neither do I” (or their alternatives) to say that you have something in common with somebody. (I am a teacher. So am I. Don’t say: “So I am.”)
So/neither + Auxiliary + Subject
So/neither + Auxiliary (from the start-up sentence; if there’s no auxiliary, just a normal, lexical verb, use “do/did”) + Person (can be any, not only “I”)
“So” is used in a positive sentence and “neither” is used in a negative sentence (take a look at the examples below).
The auxiliary verb needs to agree with the verb tens in the original statement. The auxiliary you use in your response depends on the tense used in the sentence you are responding to. Take a look at the second example; “I went to the cinema last Sunday.” – Past Tense. The auxiliary in your response has to be in Past Tense as well (“So did I”).
I love English.
So do I.
I love English too.
I went to the cinema last Sunday.
So did I.
I went to the cinema too.
I am not American.
Neither am I. (“Not me either”/”Me neither” is more informal and much more common)
I can’t fly.
Neither can I. (“Not me either”/”Me neither” is more informal and much more common)
I have been to Vietnam.
So have I. (or “Me, too” is used very often in this situation)
I wouldn’t like to go to there.
Neither would I. (“Not me either/Me neither” is more informal and much more common)