よう (and
う) are grammar constructions that function in a similar way to
ましょう, or to be more precise,
ましょう is the polite (conjugated) form of these auxiliary verbs. They are used as a casual way of expressing ‘let’s’ when making suggestions, or ‘I shall’ when making statements.
よう is only one of the potential forms of this grammar point, and can actually be divided into two separate 助動詞（じょどうし） (auxiliary verbs),
う (used with
う-Verbs), and
よう (used with
る-Verbs,
する, and
くる).
When attaching
う to
う-Verbs, simply replace the last kana with the equivalent お sound kana from within the same column (く becomes こ, う becomes お, す becomes そ, etc.), before adding
う. For
る-Verbs, replace る with
よう.
As with many other verb conjugations,
する and
くる behave in a slightly different way to regular
う-Verbs and
る-Verbs.
する will become し
よう, while
くる will become こ
よう.
The type of statements that
う, and
よう can be used for in Japanese are usually divided into three categories.
推量（すいりょう） - Expresses guesses (has a similar meaning to
だろう, which itself is the
よう form of
だ)
意志（いし） - Expresses intentions (has a similar meaning to
～するつもりだ)
勧誘（かんゆう） - Expresses invitations or encouragement (has a similar meaning to 誘（さそ）う ‘to invite’)
Although よう is used with る-Verbs, する, and くる, it should not be confused with ようだ, another auxiliary verb that is used when describing likeness (in a similar way to そうだ).