Nobody surfs anonymously, but you can try to do it with a bit more privacy.
Some while back I installed Tor, which makes a person's browsing hard-to-impossible to track. I had read about it years ago, but it took a while (i.e., some free time) to get around to installing it. The version I installed seems to arrange its own settings--convenient!--but it is slower than using my main browser, OmniWeb. As a result, I don't use Tor often, thus favoring convenience over privacy.
That said, in light of recent stories about the US, Canadian, and other governments spying on everything and everyone, to differing degrees of scope and detail, I've become a bit more sensitive. It is difficult to completely break free of Google, but the web search engine DuckDuckGo claims to keep your searches anonymous.
Instead of Gmail or Microsoft's Hotmail, try Hushmail, although as a free service its storage amount and web-login-only conditions are a bit discouraging; it has paid versions that offer more convenience. I suspect Fastmail, a paid email service based in Australia is private as well, but I don't know.
At some point I might get around to setting up PGP. It has always been on my list, just rather low.