Must, have to, should (obligation)

obligation / necessity: have to / must (infinitive)
Must and have to have a very similar meaning.
Have to is more common for general, external obligations, for example rules and laws.
Must is more common for specific or personal obligations.
Compare: I have to wear a shirt and tie at work (It's the rule in this company). I must buy a new shirt - this one is too old now (It's my own decision).
Have to is a normal verb and it exists in all tenses.
Must is a modal verb. The only forms are must and mustn't.
You can also use have to or must for strong recommendations. You have to/must see that film. It's fantastic.

no obligation / necessity: don't have to. ---- prohibition: mustn't (+ infinitive)
Don't have to and mustn't are completely different.
Compare: You mustn't drive along this street (It's prohibited, against the law). You don't have to drive - We can get a train (You can drive if you want to but it's not necessary/obligatory).
You can often use can't or not allowed to instead of mustn't.

advice or opinion: should/shouldn't (+infinitive)
Should is not as strong as must/have to. We use it to say if we think something is the right or wrong thing to do.
Should is a modal verb. The only form are should or shouldn't.
You can also use ought to and ought not to instead of should/shouldn't.