I have a serious doubt...is it correct to say: `Neither me nor my techer play tennis or smoke.`?
Correct would be:
"Neither I nor my teacher plays tennis or smokes."
But this sounds very formal, and you wouldn't ordinarily be so formal when you're talking about tennis and smoking.
My teacher doesn't smoke or play tennis, and neither do I.
Incidentally, are you Hispanic? You've used the word "doubt" where Anglos would say "question." It's a usage I've often heard in Latin America, and it's not really good idiomatic English.