Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
This only states that the Federal government shall not pass laws which limit the right of the individual citizen to express themselves freely. Simply put: you can say what you like. What it does not do, is take away the absolute right to prohibit speech, as the oft used example of shouting "Fire!" in a crowded theater illustrates. In essence, you are free to speak your mind, as long as that speech does not directly endanger the welfare of others.
This broad scope of speech means that we are subject to dealing with people who are more than willing to fill the air with all manner of tripe: homophobia, racism, antisemitism, abject ignorance, conspiracy theories, and the like. These things may make the blood of decent Americans boil, but it is intrinsically important that people holding such socially contrary views be allowed their freedom to express them, for that is the price for our freedom of speech. We cannot, of good conscience, suppress the right of others to express themselves, then claim it solely as our own. To work, freedom of speech must apply equally to all Americans.
What many fail to realize, when it comes to freedom of speech, is that the speech is never really free. You may say anything you wish, within reason. In return, those who have heard you may say anything they wish, including professing outrage at what you said. If you accept that the First Amendment guarantees your right to speak your mind, you must tacitly accept that it also provides that right to others, especially where they want to speak their mind about you. More importantly than that, if others choose to take actions based upon what you said, unless they are civilly or criminally punishable acts, they have every right to do so.
The First Amendment's ultimate goal, was to ensure that the people could speak out against the government, without fear of reprisal. The Founding Fathers wanted the Federal Government to face the music, and be forced to listen to voices of dissent, even where those voices of dissent spoke from ignorance or prejudice. It was important to them that the voice of the American people never be capriciously silenced by a government in the throes of wreaking havoc with civil liberties and the law. That same idea was implicit in all manner of speech, whereby those who could raise their voice to expound one point of view, could not simply deny those who would oppose them a means of counterpoint.
So it happens that, when someone expresses an opinion that the public finds rude, insensitive, racist, classicist, offensive, or otherwise, there is recourse to call out these statements, exposing them as such, and allowing the rest of the citizenry a chance for rebuttal, through whatever means are at their disposal. Far from being injurious to the right of freedom of speech of the initial speaker, it is, in fact, the successful application of the principle of free speech, wherein others have the right to refute what is said by someone else.
Those who operate in the mass media (television, radio, blogging, newspapers) are not entitled to some superior form of First Amendment protection, merely because of the medium in which they operate. After all, the company that hires them is free to fire them or not renew their show; free speech is not compromised because their position is at the behest of someone else, and they are subject to other strictures. A company may seek to save its reputation, which affects the advertising dollars it reaps, by removing someone who has made controversial statements. Of course, in our current society, they are more like to hire someone because they make controversial statements, in order to attract listeners/readers.
Those who claim censorship because their statements lead to their downfall, are better off looking to the question of why their statements caused such ire in the public realm. They might discover that the reason for the outcry against them may have something to do with their narrow-minded view of society, and their lack of a true connection to humanity. Simply because a person has an opinion, does not mean the opinion is relevant, factual, or important. That someone has access to a public forum to disseminate their views, does not automatically imbue them with the gift of truth. It is as possible now, as it always has been, to open one's mouth and look the fool, rather than keeping it shut to avoid the appearance. The most important facet of freedom of speech, is knowing when not to speak.