Note that 'Me too' can only refer back to statements that are grammatically positive. For example, you cannot respond to 'I don't know' with 'Me too'. For grammatically negative statements, the response is 'Me neither'.
And in answer to your question, I think it is preferable to use the echo formula 'So do I', 'So can I', 'So will I' and so on, or the equivalent negative responses.
In some contexts, 'Me too!' can sound too informal or even childish. It also cannot be used with any subject other than the first person, whereas 'So/Neither/Nor + auxiliary + subject' can be used for all persons : you, he, she, we and they.