VPNs can be purchased with BTC. Mullvad is a provider that is relatively trusted, and you can use it with Wireguard, can even do multi-hop.
You probably don't want to connect with a home cable connection. Break into other WiFi or use public WiFi hotspots. Preferably never the same one twice.
Compromised machines can be a mixed bag and require some actual work, but people definitely use them as additional routes in their chain of proxies. You could also use cryptocurrency to buy VPSes to use as VPNs potentially, which offer usefully different privacy characteristics than VPNs as a service.
Even if you get everything else right, the worst enemy will always be you. You need to regularly cycle identities, never reusing them or linking them back to previous identities. If what you're trying to accomplish does not make this possible you most likely are not going to be able to stay anonymous forever.
This is basically the same advice you'd give to someone trying to commit crimes. Well, if you consider any potential adversary to be a threat, it's pretty much the same approach. I think it's wild overkill for most folks because most folks would never have even nearly good enough opsec for this to matter - I know for a fact that my opsec is simply not going to be good enough, I'm just not that organized. For most of us, the best approach is to just not give three letter agencies any reasons to want to destroy our lives. It's easy if all you want to do on the internet is listen to music and talk about programming.
Still, despite the fact that I don't really have any need for this, I find it to be a really fun thought experiment. And there's some pretty good talks about it, too.