VPN - Virtual Private Network. You create an encrypted connection to a remote server which is the box that is logged. Some VPN services keep no logs. Some do. Now, say they only keep logs for 24 Hours. That's probably not enough time for the government to notice anything you are up to... but the log exists (so it's possible).
Tor - The Onion Router. This is a kinda weird thing. It is and isn't private. The traffic is encrypted while traveling on the Tor network, but the end-points are not. But that's not the real benefit of Tor. Your computer talks to dozens of computers before it gets to the exit node, which is where the final server sees the traffic originating from. I'm not 100% on this, but I don't think the exit node can differentiate traffic from different Tor clients. But your connection can still be analyzed and a 'profile' can be built about your online identity. FYI, Tor started as a government project. So, it is what it is.
And here's PRISM v. Tor straight from the horse's mouth.
There are also
online anonymizers. These work with HTTP and HTTPS. Some are free, some are not.
Don't forget about
proxies. These are inbetween systems. Some are used to speed up traffic, or to reduce bandwidth, but they also can provide another layer of anonymity. So you talk to the proxy, and the proxy talks to the server you want to talk to. Kinda like that telephone game when you were a kid. Again, there are free proxies you can use, but you need to be careful because they could be set up to monitor traffic. So you could, say, grab four of the public anonymous proxies, and use a utility such as proxychains to tie them together.
And finally (there's a lot more really),
don't be unique. Check out the
panopticlick.eff.org site for a demo. Basically, you don't want your web-browser to be so unique that you are the only one using it. So maybe disable javascript (yes).
Okay, that should be enough to get you started. Other members have mentioned methods on working with cell phone specific technology.
Really, when it comes down to it, security is something that is never absolute, and must be done in layers. So using Tor alone with the bundled (if you went that route) TorBrowser is a pretty good step towards anonymity. Using your regular browser (non-unique) with a VPN routing all following traffic through three or four highly anonymous proxies in different countries would be another good step. Using TorBrowser, over Tor, going through a VPN and then a few proxies would be even better (though, I bet you'd get incredibly awful throughput).
And, just to play devil's advocate, think about who you are 'up against'. This is the NSA. The NSA is constantly sponsoring new crypto systems (which, I'd wager, they vastly improve upon and use themselves). [common examples: AES, SHA] Remember Stuxnet? Weaponized malware straight from the USA. And I seem to remember a few years back hearing about a massive server cluster warehouse that was run by the NSA (I think, it has been a while).
). hopefully these doe what youd like 
