What part of speech is ought to?

Answer 1

Ought is a modal auxiliary verb. The "to" is part of an infinitive (to + base form of a verb).

ought to go

ought to have

ought to see

Answer 2

It's a contraction for should have. In a sentence, it's used as an auxiliary verb.

Answer 3

Shouldn't is a contraction of should (modal verb) and not (adverb).

Answer 4

Can and should are modal auxiliary verb.