The arguments are still familiar today. They should be as Milton lays down what we would think of in America as First Amendment freedoms. The sad thing is that we'd even need to argue this anymore, but there is some part of humanity that fears free expression.
He starts with three basic arguments: 1) 'the inventors of it be those whom ye will be loathe to own' (the censors will not be good people), 2) the order will do nothing to stop 'scandalous, seditious, and libellous books' and 3):
Last, that it will be primely to the discouragement of all learning, and the stop of truth, not only by disexercising, and blunting, our abilities in what we know already, but by hindering and cropping the discovery that might be yet further made both in religious and civil wisdom.This is a huge point, still relevant. If you restrict the scope in which people can argue, you will make it impossible to learn new things. Later on he expands on this point by comparing Jesus, the savior, to Truth itself. He says that it deceivers rent that truth apart into a thousand pieces and scattered them to the winds. That search still continues, he says, but will be stopped if the search is censored.
Milton also notes that the censorship will surely spread:
If we think to regulate printing, thereby to rectify manners, we must regulate all recreation and pastimes, all that is delightful to man. No music must be heard, no song be set or sung, but what is grave and Doric. There must be licensing dancers, that no gesture, motion or deportment be taught our youth but what by their allowance shall be honest;...Once we accept the logic of broad censorship, it grows and grows. Our current society has settled this to some extent by simply labeling some writings (music, movies, etc.) so that people have an idea of what to expect. Parents can (mostly) keep adult materials away from kids. Violent materials are clearly labeled. Religious writings are free, with some notable and important exceptions. Political writings aren't as free as they should be, but in general, opposing viewpoints are allowed.
I wish we had read this whole piece, or at least excerpts, back in high school.