This is one of those questions where the answer is "if you have to ask, you are definitely going to screw it up."
Anyway, as mentioned above, a starting point is a good VPN plus TOR.
A good VPN is harder to find than you might expect, and of less effectiveness than you might believe. The problem with VPNs is that you normally have no idea what they really do. Do they keep logs or not (in reality, not just in their P.R. pitch), are they actually independent of governments and companies interested in your information, etc? Definitely avoid the "free" ones, at least, and consider one that's based in a country with strong privacy laws.
Then there's TOR. Governments around the world are intensely interested in piercing TOR's security (ironically, it was developed by the U.S. Navy). They have clearly compromised it in the past, and likely may still be doing so. But it likely provides at least some protection.
Then there's the rest of "operational security," in which one naive mistake or slip-up will reveal who you are regardless of the above.
But true, impenetrable anonymity? You aren't going to achieve it.
