The book is much more concerned with the idea of freedom of expression than with any of the others. In this book, we see that people in this society really have no freedom of expression. Bradbury implies that this lack is one of the major problems of the society. Because they can't talk about things, they don't even bother with thinking.
In addition, legal rights have been lost. There seems to be no issue of having a trial or anything. If your neighbor says you've got books, your house is burned.
As far as equality, he seems to be saying that the desire for equality helped to cause all of these problems.
I'll center on rights for equality in Fahrenheit 451. If the novel reveals anything about equality that suggests people should be equal, it may be just that someone who disagrees with the status quo and the firemen and the government should be treated the same as someone who agrees. Clarisse and the old woman who dies with her books are not treated as equals.
For the most part, though, the novel suggests that the desire to make everyone equal, at least intellectually, is a negative. According to Beatty, books were originally banned because special interest groups, namely minorities of all kinds, complained every time something offended them. To make every book inoffensive is, of course, to make them all equal. And that is a bad thing.
Equal treatement under the law and the equal right to free speech are essential, but the watering down of intellectual thought in the interest of not offending anyone, is undesirable.
The government in the novel wants everyone to think alike, and wants everyone to be simplistic. Books that disagree can change this.