Willingness, obligation and necessity

Subject + must + verb

When we use "Must" we refer mainly to an urgent obligation:
I must go to bed earlier;
You mustn’t smoke in the hospital.

Subject + have to + verb

"Have to" also indicates an obligation but less strong » (a necessity)
Bill doesn’t have to go to work today.

Subject  + should/ought to + verb

"should" or "ought to" are used for giving someone advice.
I think you should go see your doctor.
I think you ought to try a different approach.

Subject + had better + verb

"Had better" indicates what is convenient or should be done in a current situation or in the future (never for the past).  Sometimes it is used to give warnings.  It is very similar to the meaning of "Should" but in a stronger tone:
You’d better practice before the final audition.

Subject + am/are/is supposed to + verb

"Be supposed to" indicates that someone expects something to happen or to be fulfilled.:
You are not supposed to be here