One way to protect your privacy when surfing the web is to use "incognito browsing", which will prevent others from tracing your online browsing activities. Using this option ensures that your browser is not collecting cookies or a cache that are usually stored during a surfing session.
The About Technology article Incognito Browsing has instructions on how to configure various browsers. Besides Chrome, Firefox, Internet Explorer and Safari, it includes set-up for Flock 3, Opera, iPads (Perfect and Atomic Web Browsers) and PrivateWeb for iPhone and iPods. For Android users, besides Chrome there are other free incognito browers, such as InBrowser.
Note: Use of a private browsing mode means that you will not be able to use the "Remember Me" feature for two-step verification. See the article Troubleshoot the Remember Me feature in Two-Step Verification. for more about "Remember Me".
For a short step-by-step how-to for the most popular browsers, visit Private Browing - How to Surf Incognito. For most, you can activate it with the touch of a couple keys. For example:
OSX: (UP ARROW)+⌘+P
OSX: ⌘+ Shift + N
⌘ is synonymous with the Command key