The volitional form is used in the same way as the -ましょう form, but in casual and familiar speech.
Its most common use is equivalent to “let’s …..” in English.
手伝おうか？
うん、手伝おうね。
Shall we help them?
Yes, let’s.
Used in a question, it is equivalent to “shall we ….?”
行こうか？
Shall we go?
It is often addressed to the speaker’s group (we), but can also just refer to the speaker him/herself.
やってみようか？
Shall I give it a try?