ましょうLet's, Shall we (Polite volitional)

ましょう and Verb[よう] are both volitional forms that mean 'Let's (do something).' The difference is tone: ましょう is polite, while Verb[よう] is casual. For example, 行きましょう (polite) and 行こう (casual) both mean 'Let's go.' The casual Verb[よう] form is also used in other grammar patterns to express intent, such as 行こうと思う ('I think I'll go' / 'I'm planning to go').

ましょう and ませんか are both polite expressions based on the ます form that can be used to make suggestions or invitations. ましょう is more assertive and usually implies the speaker and listener will do something together (e.g., 食べましょう = 'Let's eat'). In contrast, ませんか is more of an invitation directed at the listener (e.g., 食べませんか = 'Won't you eat [some]?').

Both ～ましょうか and ましょう are polite expressions used to suggest or propose an action. ましょう is more assertive, similar to 'Let's (do this)' in English. In contrast, ～ましょうか is softer and seeks the listener's approval or agreement, like 'Shall I...?' or 'Shall we...?' ましょう can also function as a gentle command, such as when a teacher says 'Let's be quiet' to mean 'Be quiet.'