Grammar JLPT Level N5 - Lesson 17

As covered in our ましょう lesson, ましょう is one of the conjugations of the auxiliary verb ます, and is a common (polite) way to suggest that something be done 'together'.
Adding the particle か to this expression simply changes the nuance from 'let's', to 'shall we?', and is more of a direct question than ましょう by itself.
Like ましょう, this expression can be added to the ます stem form of any verb.

• レストランまで車(くるま)で行(い)きましょうか。
  Shall we go to the restaurant by car?
• 今夜(こんや)走(はし)りましょうか。
  Shall we run tonight?

One thing to note with these examples is that they are 'suggestions', not 'invitations'. In Japanese there is a slight difference between the two, in the same way that there is in English. If you want to invite someone to do something, the most natural way would be to use ませんか, a phrase that is similar to ましょうか.

• 一緒(いっしょ)にイタリアに行(い)きませんか？
  Won't you go to Italy with me?
• 一緒(いっしょ)にイタリアに行(い)きましょうか。
  Shall we go to Italy?

The difference here is that 'won't you do (A)?' has a stronger nuance of being a direct invitation, while ましょうか is closer to just 'let's do (A)!'