〜ましょう
let's (polite)
Verb ます-stem + ましょう
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Replace ます with ましょう to suggest doing something together: 行きましょう, "let's go"; 始めましょう, "let's start".
It's upbeat and assumes the other person is on board, so it's used when the plan is more or less settled, or by whoever's in charge (a teacher: "let's open our books").
To actually invite someone, where they might say no, 〜ませんか is softer. That's the next point.
The casual version is the volitional form: 行こう, 食べよう. You'll hear it constantly between friends and in anime.
Register: Polite. Casual: the volitional form, 行こう, 食べよう.
Examples
もう遅いですから、帰りましょう。
It's late, so let's go home.
一緒に食べましょう。
Let's eat together.
では、授業を始めましょう。
Right, let's begin the lesson.
駅の前で会いましょう。
Let's meet in front of the station.
ちょっと休みましょう。
Let's take a short break.