The grammar patterns よう and う are the informal counterparts to ましょう. In fact, ましょう is simply the polite conjugated version of these plain-form auxiliary verbs. They are used to casually suggest an action ('let's...'), or to state one's own intention ('I will...').

This grammar point has two forms depending on the verb type. The auxiliary verb う is used with Godan verbs (う-Verbs), while よう is used with Ichidan verbs (る-Verbs), as well as the irregular verbs する and くる.

To conjugate a Godan verb (う-Verb), change the final kana from the 'u' vowel sound to the corresponding 'o' vowel sound in the same column (e.g., つ becomes と, む becomes も, ぶ becomes ぼ), then add う. For Ichidan verbs (る-Verbs), just replace the final る with よう.

• お母(かあ)さんのプレゼントを買(か)おう！
  Let's buy a present for Mom!
• 寒(さむ)いから、窓(まど)を閉(し)めよう。
  It's cold, so let's close the window.

Similar to other conjugation patterns, the irregular verbs する and くる have their own unique forms. する changes to しよう, and くる changes to こよう.

• 部屋(へや)が汚(きたな)いね。私(わたし)が掃除(そうじ)しようか？
  The room is messy, isn't it? Shall I clean it?
• もう遅(おそ)いから、また来週(らいしゅう)こよう。
  It's late now, so let's come again next week.

In Japanese, the use of う and よう can generally be classified into three main functions.

推量(すいりょう) - Conjecture or guessing (carries a meaning close to だろう, which is the volitional form of だ)

意志(いし) - Expressing one's will or intention (similar in meaning to ～するつもりだ)

勧誘(かんゆう) - Making an invitation or suggestion (comparable in meaning to the verb 誘(さそ)う, 'to invite')