anarchy doesn't have a goal. anarchy is, perhaps among other things, a way of being in the world. the goal of anarchists, or anarchism, is to get to anarchy, and/or to practice anarchy in our day to day lives. (some anarchists think that anarchy is unrealizable right now, because of the constraints and authorities surrounding us all the time, and some people think that anarchy exists in our lives in moments, whenever we ignore or effectively move around those constraints and authorities). one thing that i think i can say for all anarchists is that we think laws are part of the problem. they are created by the power structure we consider invalid, to support invalid agendas. so yes, we don't care about (or in some cases actively support) breaking laws. short term goals of some anarchists are reformist, to make people's lives better here and now (and what they mean by that will vary based on what kind of anarchist they are). other anarchists don't have any hope that anything they do will bring about a different world, and act to make their own lives and the lives of their friends better (and again, what they mean by that will depend).
there are various ways that anarchists think that anarchy could be more prevalent. some anarchists think that civilization has to go, because the things that civilization requires (centralization, etc) preclude the kinds of social relations that foster anarchy. other anarchists think that we can have a world roughly like the one we have now, but without politicians or bosses, with people making all the decisions about their lives in meetings with each other, so that no one gets too much power, and no one doesn't have a say in decision making.
obviously, then, based on that, different anarchists have different ideas about what an anarchist society would look like. from small-ish self-sustaining groups (for example as hunter/gatherer groups used to be) to cities but very differently organized. other anarchists aren't that interested in envisioning a future anarchist society (or societies), just knowing that they will be very different, in ways we can't imagine because of our limitations from having grown up in the current situation.
i will answer your final question differently from how i responded to the first parts: as far as i'm concerned, part of the problem of how we are in the world today is by thinking that "the natural world" is distinct from us-as-humans. we are part of a whole, and my goals are that we remember/learn that. but there are anarchists who are transhumanists, and i don't know how they would answer that question. i think they want off the planet? unclear.