I'm going to make an assumption that, for your purposes, you most likely don't have anything to worry about, but still:
Does participating in an online community about anarchism out you on a watchlist? There's no way to know what precisely constitutes a "watchlist" or what criteria US intelligence agencies use to populate them, but we do know that the DOD considers anarchists an "extremist ideology" that constitutes a potential "internal threat." Public communications and documents from US agencies indicate that "anarchist extremism" and/or "anarchist violent extremism" is a threat that is being monitored by law enforcement and intelligence agencies.
To address the question of violence, we also know that certain methods of surveillance by the US government are notoriously broad, and are not predicated on the commission of a crime, so it's not unlikely that simple suspicions of anarchist extremism could land your data in an NSA database somewhere. However, your data on an NSA watchlist is very unlikely to get you arrested if all you do is talk about the ideology and don't make anything that constitutes a threat of harm. And you're probably in that database anyway for some other reason.
Can you get arrested/charged for crimes? Not to be hyperbolic, but police can arrest you and charge you with crimes for any reason at any time. There's nothing you can do that guarantees you won't ever be arrested. That being said, being arrested and charged does not mean you'll be convicted, and police don't have infinite resources to just arrest anyone and everyone forever.
Still, are there examples of anarchists who have been spuriously arrested and charged with crimes despite being nonviolent? Yes. I've been arrested for nonviolent, legal political protest. I wasn't breaking any laws. I spent a day in jail on a made up charge, then they released me and never pursued the charge in court. Just a scare tactic. That's not an unusual story.
For a more extreme and recent example, anarchists in Atlanta are being charged with domestic terrorism, and one of them was extrajudicially murdered, for their nonviolent protests to stop cop city. Organizers for the legal, nonprofit bail fund in Atlanta were also arrested and charged spuriously with money laundering and charity fraud in a related case.
Two things about those incidents, though. One, these arrests are by local police with a grudge, and so I doubt it's related to a federal watchlist. Two, the victims are actively engaged in organizing for a political protest movement, not just chatting about Bakunin and Proudhon on a forum.
I think you should get involved, talk, and learn more about anarchism, socialism, communism, any stripe of anti-capitalism. I'm confident that you can learn about it and chat about it safely, regardless of whether your data ends up in an NSA database. But you should be aware that getting involved in IRL organizing on the left, even nonviolently (as it almost always is), isn't safe. It's absolutely, 100% worth doing, but political resistance has never been safe, and part of a good education is learning how to protect yourself.
Just because nobody's said this yet and it needs to be said: the people who know the answer won't tell you, and the rest of us don't know for sure, we're only guessing.
You can be placed on a watch list for many things, even seemingly mundane things. It just depends on which three letter agency is feeling the most paranoid at the given moment.
You can’t get arrested or charged for exercising your right to free speech in the US. What you can get arrested and charged for is plotting to overthrow the government. 18 U.S. CODE § 2381
Discussing the benefits of anarchism or generally advocating for a move in that direction, without planning or participating in a coupé, is not a crime. Coming up with a plan to violently revolt and destroy the government is a crime. There are nonviolent ways of achieving political change.
It'll probably slap you on a watchlist, but that's not really a huge accomplishment. As far as getting charged, it depends on where you live and what you say. If you start talking about blowing up buildings or something, well, I hope you have a good lawyer on retainer before the feds hunt you down.
I'm thinking watchlists are a thing of the past
They're a tool that people could use to generate leads on cases. Now there's AI. You probably have a full dossier of all your interests, and your name would come up for any tangential case
Does it "put you on a watchlist"? Well, it depends on just what you mean by "a watchlist". If you mean any list anyone has then yes, I'm sure there is someone making a list of who they saw participate in a community about anarchism. Probably some other member of the anarchist community who is looking for like-minded friends. If you meant some US government list that is used to persecute people and deny them jobs -- we did that up until about 1975, but it's not something we do today. If you meant some US government list that is secret, but might be used to consider whether to give someone a security clearance then the answer is maybe -- after all, the list is secret, so we wouldn't KNOW if it exists.
"Can you get arrested/charged for crimes?" No. Discussing anarchism is NOT a crime. Now, if you have a chat with your anarchist friends about the best way to build a bomb and about how you plan to use a bomb on October 7 to blow up the President or some Supreme Court justice, then THAT is a criminal conspiracy, and you should expect to be hunted down and arrested. But if you are just discussing policy the First Amendment guarantees your right to free speech and free association. And while the enforcement of that has been spotty over this history of this country, at the moment we don't put people in jail for joining an online community.
Is believing in Anarchism illegal in the USA? I don't think so. The only way in the USA to get on a watchlist is to be either a high felony criminal or a terrorist. Neither the CIA nor the FBI have the time and resources to monitor everyone at the same time.
technically signing up for any service gets you put on a list, and there are very few lists anyone has - including handwritten ones in your kitchen drawer - that the government/police couldn't access if they really wanted to.
Privacy, safety, security are conventions and habits of politeness mostly.
Really the main protection is that there are much bigger fish to fry than people reading internet posts about politics, and even within anarcho-social circles, there are bigger players with more concerning activities than you.
As long as the Gmen don't start using cheap bots to filter their search parameters for "threatening" words and the country remains stable you should be safe.