The Internet is the last bastion of the free exchange of ideas in the world. Of course the First Amendment right to free speech applies to the Internet as much as possible. Just like real life, one cannot make death threats to another person nor can someone say "fire" in a crowded chat room. There are limits to free speech just as in normal communication.
Yes, the First Amendment right free speech should -- and does -- apply to the Internet. If you read the First Amendment (and it can just be a skim, doesn't have to be an in depth examination), you'll find that it does not specifically mention the medium of speech protected. It protects all speech.
The right to free speech should apply to the Internet, but of course that would only be for websites that are based in the United States. We also need to realize that "freedom of speech" is a limited freedom. Some people believe that freedom of speech means they can say whatever they like, but this isn't the case. There are limits. Some things are not allowed, like lying to ruin someone's reputation and hate speech meant to incite others.
Just as you can not libel or slander someone in real life without getting sued, then the same rules should apply on line. That means you should be able to speak your truth unless it is not true or unless it is so inflammatory that it is starting something that one would not instigate in real life.
The freedom of speech that we have in the United States should apply to the Internet, but, of course, only for United States citizens. We need to respect the rights of other countries, such as China, that have their own restrictions and limitations on what their citizens can access. For us, the Internet is an extension of media, and we have the right to say what we want online, as long as it is within the limits given to us by our country. For example, freedom of speech does not apply to things like inciting others.