Does the first amendment protect students at school?

Well, there are difference categories of the first amendment.

It is argued that the First Amendment rights of students are allowed to be limited more because students are in a social contract with the school; this means students must give up some rights for the common good in exchange for an education and all that comes with it.

The case Tinker v. Des Moines (a case about free speech), the Supreme Court established that students "do not shed their First Amendment rights at the schoolhouse gate". Yeah, sure, we have first amendment rights, but they're very limited. For example, you may have heard of the "Bong hits for Jesus" case (Morse v. Frederick) where they limited the kid's speech. When it comes to speech, they're also very strict when it comes to obscenity because it disrupts the learning process.

As for religion, there are many cases about it. (Try firstamendmentcenter.org). One example is Santa Fe v. Doe. The students held a vote about whether or not they wanted someone to lead a prayer on the loudspeaker at the football game at their public school. They held a vote and decided to and chose someone to lead it, but since it was public, Supreme Court found that it was a violation of the establishment clause.

There isn't a ton of press in schools that doesn't fall under freedom of speech, but one example is Hazelwood v. Kuhlmeier when they allowed the school to limit the speech.

(not addressed in answer: right to peaceably assemble and right to petition the government.)