Should and ought have very similar meanings. In fact, they can often replace each other.

We should work hard, shouldn’t we? OR We ought to work hard, shouldn’t we?

Both should and ought are used to talk about obligation and duty. They are also used to say what we think it is right for people to do. Note that should is more common than ought.

You should have a word with him. OR You ought to have a word with him.

Should and ought are not used in polite requests.

Could you move a bit? (NOT Should you move a bit?)

Logical possibility
Both should and ought can be used to talk about logical possibility.

I have bought two dozen apples – that should / ought to be enough.
That should be John knocking at the door. OR That ought to be John knocking at the door.

Notes

Should is followed by an infinitive without to. Ought is followed by an infinitive with to.

We should help the needy. (NOT We should to help the needy.)
We ought to help the needy. (NOT We ought help the needy.)

Must and should/ought

Must have similar meanings to should and ought, but it is stronger.
Must shows great confidence that something will happen. Ought and should express less confidence.

The teacher said that the students must learn the lesson. (This is kind of an order which is likely to be obeyed.)
The teacher said that the students should / ought to learn the lesson. (This is kind of a piece of advice which may or may not be followed.)

That must be John. (= I am sure that it is John.)
That should / ought to be John. (= I think that is John.)