Should, Ought to, Had Better

Should and ought to are used to give advice or making recommendation. They have the same meaning (it is advisable to do something).
For examples:
• 'This soup is too salty!' 'You should / ought to send it back.'
• You'll catch cold if you go out like that. I think you should / ought to take a hat.

Should and ought to are also used to shows responsibility or duty.
For examples:
• People should / ought to be warned (= passive) of the danger of swimming off this beach.
• I should / ought to visit my parents more often.

We can use 'had better' instead of 'should / ought to' to say that we think it would be sensible or advisable to do something.
For examples:
 • If you're not well, you should / ought to ask Ann to go instead, (or ...you'd better ask...)
• You should / ought to have caught a later train, (not You had better have...)
• I don't think parents should / ought to give their children sweets, (not ...had better...)

We can use (be) supposed to instead of should / ought to to talk about an obligation to do something. It is commonly used in spoken English to express a' less strong obligation:
For examples:
• I'm supposed to be there at 10.00.
• The work was supposed to start last week.

We use (be) supposed to when we report what many people think is true:
For example:
• Eating sweets is supposed to be bad for your teeth, (not ...should be bad for...)

Reference

Martin Hewings (1999). Advanced Grammar in Use. A Self-Study reference and practice book for advanced learners of English