Have to is often used for rules.
You have to work hard. (it's a law) (Dovere, non obbligatorio)
Must is often used when the obligation comes from the speaker. Must is never followed by to.
I must remember it. (I think this) (Obbligo, devo assolutamente ricordarmi)
Mustn't means it is prohibited.
You mustn't waste your money
Don't have to means you have a choice.
You don't have to eat the food on an aeroplane(it's not necessary but you can if you want to)
You should start early. (it's a good idea)
You shouldn't stay late. (it isn't a good idea)