There are various forms to express both agreement and disagreement and there is of course a difference between them.
Some forms of agreement:
- The one you mentioned ( me too) and it is used to indicate that the person adapts the same attitude as the speaker, feels the same way or does the same action.
- " Yes I couldn’t agree more."
- " Exactly."
- " Yes, I quite agree."
- " That’s quite true."
- " I think you are absolutely right."
'Neither I' is used when we want to show that we feel the same way as another person, or have performed the same action. So it is a form of agreeing as well ,but with negative sentences.
Some forms of disagreements on the other hand:
- " I can’t agree with you there."
- " I am afraid I disagree with you."
- " Perhaps,but I don't think that ....."
- " You can’t be serious."