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LingakuLingaku[v5] 〜う → 〜おう: 買う → 買おう、行く → 行こう、話す → 話そう、待つ → 待とう、飲む → 飲もう、遊ぶ → 遊ぼう、帰る → 帰ろう、泳ぐ → 泳ごう [v1] 〜る → 〜よう: 食べる → 食べよう、見る → 見よう [する] → しよう [来る] → 来よう [verb ます-stem] + ましょう
The volitional form says "let's" when you suggest doing something together, and "I'll" when you decide on the spot to do something yourself. A る-verb swaps る for よう (たべる食べる → たべよう食べよう), a う-verb changes its last sound from -u to -ou (いく行く → いこう行こう), する becomes しよう and くる来る becomes 来よう.
This is the plain form, for friends and family. With anyone else, use the ます-stem plus ましょう.
Only る-verbs take よう. A う-verb that happens to end in る still changes る to ろう.
To offer to be the one who does it, name yourself with が. が picks you out from everyone present.
A bare volitional is a decision made on the spot, often said half to yourself, and it speaks only for the speaker, or for the speaker and listener together. To tell someone about a plan, add とおもう思う; とおもっている思っている is a plan you have had for a while.
A bare volitional can't report someone else's plan. With と思っている it can.
These go beyond "let's" and "I'll".
Adding か to ましょう makes it a question: "shall we…?" to suggest something, "shall I…?" to offer help.
ましょう assumes the other person is on board. ませんか asks whether they want to, so it is the softer invitation when you don't know the answer.
With とする, the volitional describes an attempt, or the moment just before an action.