Considering that the internet transmits your public IP address in every header you send across the internet and also contains the IP address of the destination, there is no way for you to hide what sites you visit without going through a proxy server. As far as I know, Header information in every packet is plain text and there is no way to encrypt that because if it was encrypted then no router would be able to forward your packets onto the next step in its final destination. So your browser, e-mail program, or anything else that sends and receives data through the internet is going to leave a trail for the government to potentially record. It may not lead back to you specifically, but it will lead to someone in your household or in your neighborhood that is using your wi-fi for internet access, provided you haven't locked down your wi-fi. If you have locked down your wi-fi then the government can claim it was only you, someone in your household or someone you have given your wi-fi password to, which significantly lowers their potential suspects or targets.
If you send everything you do through a proxy server with a vpn connection to the proxy, then that has a very good chance of making you mostly anonymous. However, a warrant and the cooperation of the proxy service owner might make it possible for the government to still connect the dots back to you. Also, sending everything through a proxy server with all the non-routing information encrypted (via vpn) may actually lead to you being watched more closely then if you don't.
If what you are really after is encryption of the contents of what you see and do on the internet, your best bet is probably still a VPN through a proxy server. Especially since SSL and some of the other methods for encrypting data between two end points on the internet aren't as secure as they were once thought to be. I don't know of anyone that has come up with a replacement for SSL that has been adopted by very many content providers. And even if the web browsers may have adopted some new security encryption scheme, it won't be effective until most if not all content providers also adopt and implement it.