「ましょう」is a broken sound of 「ませう」, which consists of 「ませ」＋「う」. 「ませ」：「ます」の未然形 「う」：particle indicating 'guess' - さあ、行きましょう　= Maybe we are gonna go -> Let's go - 私が行きましょう = Maybe I'm gonna go -> I will go. and you can say - 明日は雨になるでしょう。　Maybe it's gonna rain tomorrow. - もう3年になるでしょうか。 Maybe it is 3 years since then, isn't it?

you can use しょう as "will" l'll go l'll do it for you lt will rain etc... not only "let's ～"

I think that "私がいきましょう" means that I will go in behalf of you. For example, at least one of member of group should attend a meeting, but everyone hesitates. In that case, I sometime use the phrase of "私がいきましょう" even if I want not to go, too, in order to wrap up promptly the discussion who should go.

みんなさん，ありがとうございます。