"Das muss ich tun."
Translation:That is what I must do.
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I've always thought of it as the same differences in english between "do" and "make", with the phrase "What can I do for you?" - "Was kann ich für Sie tun?". In that sense you can do something for someone, but you are not making a thing, you can make homework for example, since it is not an "action" per se, but the realization of something. You can't say (as far as I know) "Hausaufgabe tun", you say "Hausaufgabe machen". If the action is unknown, you can ask with "tun": "Was hast du getan?" (What did you do?) - "Ich habe meine Hausaufgabe gemacht" (I did my homework).
'tun' for actions
'machen' for concise activities
I hope my explanation wasn't too convoluted or did not confuse you further.
"Müssen" is frequently translated as "must," so your teacher is incorrect here. In fact, I really don't see any meaningful difference in English between "must" and "have to."
"Sollen" is generally translated as "should" or perhaps "ought to." Essentially "müssen" is a stronger obligation; "müssen" shows more or less of a necessity, whereas "sollen" shows something that is simply wise to do.