Good question. "Memorize" is actually not a formal word. We use it in normal, everyday, casual English. There are alternatives, but "memorize" is the most common word, by far. But, since you asked, some other options: "Learn by heart" is another way of saying it: "The couple should've learned their vows by heart." Another option is "to get something down pat": "I've got my lines down pat, I'm ready to go on stage!" This is very informal and American; it can also be used for a skill, not just for something to be committed to memory. Like I said before, "memorize" is the best option in any context!

Your first example has a number of mistakes making it unclear. What you probably mean is: The teacher told the students to memorize the text on page 56 before (the) next class. In place of ‘memorize’ you could simply substitute ‘learn’ here. If you are talking about learning a text, that means memorizing it. There is nothing wrong with using ‘to memorize’ though. It’s natural.

Learn by rote is another way of saying memorize

In addition to Eric's suggestions, you can also use 'commit to memory'.