Vanilla Internet doesn't include anonymity in the package, LOL.
You gotta have an IP address and you have no way to hide it by any means other than using another IP address, and then you have to trust that other IP provider to hide your ass. So, it becomes a trust game, and in the largely public system the Internet is, you cannot trust random people, really.
Anonymity itself isn't even a technical problem, it is a social problem, and it shouldn't be confused with privacy. For two people to have conversation completely anonymously, they have to put their trust into something else (for example a wall in the back street or a web forum) to hide their identities from each other. And that of course jeopardizes their privacy. And vice versa, to have a completely private conversation (vis-a-vis), you have to give up your anonymity for at least a pseudonym (like with RSA shared keys).
However, with systems like Tor and other darknet tools, you might approach very "anonymous" and "private" communication indeed. Tor essentially dissolves your identity into the one of the "Tor user", by routing your message through 3 nodes, with last two agnostic of where the message came from originally and the middle one completely breaking up from the original sender, making it rather non-trivial to trace the message back to its sender. Of course the system has weaknesses, but it has an OK idea - putting message through a lot of other users as privately as possible.
But of course, however, you just put both your privacy and anonymity into hands of random CIA niggers in this case anyway. Though anonymity is more easily jeopardized in this game, I guess. "Tor user" is already too non-normie and Tor traffic is distinguishable from the regular TLS traffic on the wire.
What I want to say is that using Tor and VPN/VPS bought for bitcoins or whatever means not connected to your real identity is VERY strong at hiding your ass, at least so far. Don't shit your pants too much.
Also, the last thing, if your actions have severe repercussions IRL, CIA niggers will likely to find you anyway. It is an effort/reward ratio game for them. I mean, sure, you can run, as in be quick on your feet at using various public wifi spots with security cameras also hacked by you (like, it is pretty fictional shit already, but it might be possible for some of them), but most probably cannot hide. So, ultimately, (((they))) have an upper hand over you IRL, unless you belong to a rather enclosed criminal organization, but then you are vulnerable through other means, e.g. traitors and other shit.
OK, this post is too long.