So the best place to start is, What is Anarchism, and to narrow this down i'm going to talk about Anarchism as a philosophy, i'm not talking about the many sects of Anarchism, i'll get into that later on.
-- What is Anarchism?--
Anarchism is the opposition to all unjustified hierarchy. So what does unjustified mean? Unjustified means that there is no moral, ethical or general reason for the Hierarchy to be there, because to the anarchist, this hierarchy must exist only if there is no viable alternative to it.
So what are examples of unjustified hierarchy, and what are some possible solutions to fix these hierarchies? Capitalism is the main one, Capitalism creates wealth gaps between individuals, or to put a more Marxian spin on it, it drives a wedge between groups of people based on their Socio-economic class (how much financial, cultural and social capital they have). This system creates class conflict as there is conflict between those with Capital (bourgeois) and those without (the proletariat), now you could say those in the top of society worked hard to get there, but this isn't the case in 99% of western capitalist economies, most billionaires got their class and power from their parents, their grandparents, or even their great-grandparents, this capital had to come from some kind of exploitation, namely in the western world in regards to inherited wealth, The slave trade is the most common source of inherited capital. The solution to this would be a collectivist economy, and this is where the scary buzzword comes in, Communism. Now when Anarchists talk about Communism we aren't talking about some kind of massive state like in the USSR, China or Cuba, but instead we are talking about a system decided by the workers for the workers. The workers control the means of production and everyone gets their basic needs met, so this allows for a system that allows you to keep your home, have food, water and clothing, since these are basic needs in order for you to survive.
So what is another hierarchy that anarchists are against? The police, now i hear what you are going to say, "if we didn't have police, people would kill eachother all the time" and to that i say, if you go out into rural areas, how many policemen do you see? Maybe 1, 2 at a push, how much crime do you hear coming out of those areas? Pretty much none in the large scheme of things. Everyone in those areas knows eachother, and they know how to deal with crime when it arises, the same would happen in an Anarchist commune, because they are so close knit, everyone would have a hand in dealing with people who break the rules of the commune, it would be a much more diplomatic way of dealing with crime since you'd be bringing the judgement of the criminal back to the community they criminal has harmed rather than taking the responsibility away from the community that they have done damage to.
There are many forms of Anarchism, all are left wing, since Anarchism in its roots has been against the status quo. The status quo here being Capitalism, so it makes very little sense for Anarchism to be in favour of anything that tries to continue a system that creates unjustified hierarchies.
--Some forms of Anarchism--
Anarcho-communism is the most out-spoken form of Anarchism, Anarcho-communism calls for the Abolition of the state, wage labour, markets and everything to do with Capitalism. Anarcho-communists want to have a society based on the core principles that Peter Kropotkin laid out, the main ones being Direct Democracy and Mutual Aid. Mutual Aid in its core ideas is the idea that people should help each other out, not for monetary advancement, but instead help each other out because both parties are getting mutual benefit out of the transaction, kind of "i scratch your back, you scratch mine" basic altruism. Direct Democracy is the practice that took place in ancient greece, everyone in a community, gathers in one place to come up with new rules, changes to rules and everyone has a equal say in how the community they live in is run. This can be everything from drug legislation to trade regulations with other communes.
Mutualism aka Anarcho-Mutualism believes in markets but with a mix of collective ownership and private ownership, so basically the idea is that there is a equal mix of collective and private ownership but everything can be bought and sold on a market. There is no state, thus there is very little in the way of regulation, to me mutualism could be seen as a actually legitimate version of "Anarcho"-Capitalism in that it in theory has everything Ancaps want, it has the markets, and the limited regulation but its actually a pragmatic idea since there is still direct democracy, and there is no meme of recreational nukes.
Anarcho-Syndicalism and Anarcho-communism go hand-in-hand, most Anarcho-communists are Anarcho-syndicalists (ANSyns) and most Syndicalists are Anarcho-communists. Anarcho-syndicalists, are the old school trade and industrial unionists of Anarchism. The strive for the workers to collaborate in one big union (the IWW for example) where they can starve the government through revolutionary praxis like monkey-wrenching, wildcat strikes and occupation of workplaces, Ansyns are more Revolutionary Praxis, than Revolutionary Theory.
This includes Anarcho-naturalists, Veganarchists and to an extent Anarcho-primitivists. Green Anarchists strive for a Anarchist utopia where the workers own the means of production like in Anarcho-communist societies, but they put an emphasis on Animal welfare and the environment. They see Capitalism as the root cause of the earth dying, and they see the revolutionary workers as the root solution to fixing the issue. Green Anarchists are usually Anarcho-communists or Ansyns. Not all Green Anarchists are vegans but a vast majority are at least Vegetarian in their praxis under Capitalism.
--Anarchist Praxis under Capitalism--
Anarchism is much a way of life as it is something to strive for. David Graeber made a good point in saying that there is no point in calling yourself an Anarchist if you don't do anything, Anarchism is something that you do everyday. So everyday to be a true anarchist you need to do Anarchist praxis, Anarchist praxis isn't just blowing stuff up and punching Nazi's although those are to some degree necessary aspects of praxis but as an Anarchist you need to be part of campaign groups, you have to be vocally against the unjustified hierarchies that exist in our society. Some aspects of Anarchist praxis are finding the names of vocal nazi's and finding out where they work then telling their boss to get them fired, another one would be to open up a homeless shelter, or at least accepting a few homeless people into your home and helping them to find a stable home.
My job here writing this is not to do your work for you, my job is to put time into you discovering yourself, i don't have all the answers, and i never claim to. The job of a human being is to gather knowledge, i simply see this as a stepping stone for you to understand were me and other anarchists are coming from. Wander outside of your echo-chamber and learn what we have to say, don't look at biased media. Talk to us, as human beings, and accept that we don't have all the answers but at least accept we have an opinion and ideas that could make life a lot easier than it currently is, for everyone, not just ourselves. We are human, we are social creatures, so stop acting like Anarchists are edgy teens and accept that maybe for once, the edgy teens have a point. I'll point you in the direction of some reading material, but its up to you on whether you read it or not, i'm not you, its not up to me to force feed you information.
--Anarchist reading material--
An Anarchist FAQ by The Anarchist FAQ Editorial Collective: https://theanarchistlibrary.org/library/the-anarchist-faq-editorial-collective-an-anarchist-faq.pdf
The Conquest of Bread by Peter Kropotkin: https://theanarchistlibrary.org/library/petr-kropotkin-the-conquest-of-bread.pdf
Anarchism and other essays by Emma Goldman: https://theanarchistlibrary.org/library/emma-goldman-anarchism-and-other-essays.pdf
Debt: The First 5,000 Years by David Graeber: https://theanarchistlibrary.org/library/david-graeber-debt-the-first-five-thousand-years.pdf
Fragments of Anarchist Anthropology by David Graeber: https://theanarchistlibrary.org/library/david-graeber-fragments-of-an-anarchist-anthropology.pdf
Are you an Anarchist by David Graeber: https://theanarchistlibrary.org/library/david-graeber-are-you-an-anarchist-the-answer-may-surprise-you.pdf
Anarchism, Or The Revolutionary Movement Of The Twenty-first Century by David Graeber and Andrej Grubacic: https://theanarchistlibrary.org/library/andrej-grubacic-david-graeber-anarchism-or-the-revolutionary-movement-of-the-twenty-first-centu.pdf