sorry to say, cherry, we never learnt "either" in english like what you told.
as i know, too and either can both mean "also/as well" as adv. in the end of sentence.
the difference is: too for possitive statement but either for the negative.
---I'd like to have a rest. ---me too./I'd like, too.
---I can't swim. ---I can't, either.
in the second conversation if we use neither instead, we have to say "Neither can I."...