We nearly always say "insist on" (or "insist upon") or "insist that":
"The teacher insisted on punctuality from the students."
"The teacher insisted that students be punctual."
Notice that "insist on" is followed by a noun form ("punctuality"), while "insist that" is followed by a clause in the subjunctive mood ("students BE punctual"). A clause must have its own subject and verb. The subjunctive mood takes the verb in the infinitive form without "to" for ALL persons: I be, you be, he be, we be, they be, etc.
We also use the intransitive (followed by nothing) form, "I insist" as a polite invitation:
"Please, take my seat. I insist."
Really, this has an implied clause: "Please, take my seat. I insist (that you take it)."
And you will see this in reported situations as well:
"I didn't want to accept his expensive gift, but he insisted (that I accept it)."