First of all, you should provide a context with this kind of question.
He insisted on reading the letter - does not say whether he has eventually started to read the letter or not. It only says the he expressed his desire to do it.
"he insisted on getting start to read.", "he insisted on continuing reading." - did not get your question here. Those sentences mean: He wants to begin with reading, he wants to continue reading. Do you want some other ways to say it?