The Tor Browser is about as solid an option there is for anonymously surfing the web. If you'd rather that all internet access around the house be cloaked in the same anonymous stealth mode, however, you'll need something like the Safeplug.
Made by Pogoplug, it's a box that connects to your router and automatically uses the Tor network for all internet requests that happen in your home network. If you're not familiar, Tor (short for The Onion Router) is a networking protocol that anonymizes any data relayed across it by bouncing everyone's traffic around other users' connections. It's not 100% secure (nothing is), but it's about as close to airtight anonymity as you can get online.
To use, simply hook up Safeplug to your internet router and activate it via Pogoplug's web app. Once that's done, you're good to go, encrypting everything that comes and goes in your home network (even data from your phone), then routing it through a variety of random computers. If you'd rather not use Tor for certain browsers and machines (Tor just isn't feasible for video streaming, Flash content, and sites that need to verify your actual location, among other things), you can do it in the settings. If you'd like to help out the Tor system, you can also set the box to use your computer as a node for bouncing some of the data that passes through it.
Whether you're a hacker, a spy, or an unbathed criminal who ply your evil trade online, the Safeplug is about as simple and straightforward a device to obscure your online activities. It's available now, priced at $49.