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LingakuLingaku[verb ます-stem] + ましょう [verb ます-stem] + ましょうか
～ましょう means "let's". Take a verb in its ます-form and change ます to ましょう.
～ましょう takes it for granted that the listener is in. That makes it the form for a plan that already exists, and the standard yes to an invitation.
When you don't yet know whether the other person wants to, invite with ～ませんか. ～ましょう said to somebody who has not agreed to anything is pushy, and more so to a teacher or a boss.
With か on the end it becomes a question, and it has two readings. One is a softer "shall we?". The other is an offer to do something for the listener: "shall I?".
These go beyond "let's" between two people.
Teachers, signs and announcements use ～ましょう to tell people what to do without ordering them.
～ましょう has no negative of its own – nothing like ×行きませんしょう exists. For "let's not", use やめましょう, from やめます (to stop, to give up).
Between friends, ～ましょう is replaced by the plain volitional form: いきましょう行きましょう becomes いこう行こう, たべましょう食べましょう becomes たべよう食べよう.