You have a few options to protect the privacy of your Internet searches. You could log out of your Google account before using Google to perform a search. You could use an alternate browser to keep your searches separate from your primary browser. Or you could use a service such as Duck Duck Go. If you are set in your ways (and use Firefox as your primary browser), however, and want to search using Google like the rest of the world and want to do so without logging in and out of your account or switching browsers, there is a Firefox extension that promises to keep your Google searches anonymous.
Simply install the Searchonymous extension and go about your business. The extension does not require a restart of Firefox, and there are no settings to fiddle with. Searchonymous does its work in the background. You'll stay logged into the other Google services you use -- Gmail, Google Drive, YouTube, Picasa, and the like -- but when you arrive at Google's search page, you'll see the "Sign in" button instead of your profile pic or user name. Conveniently, the grid button that lets you quickly access other Google products remains on the page.
According to developer MorbZ's description of Searchonymous, the extension keeps your searches from being tracked by "blocking certain Google cookies (such as the PREF-ID) or generat[ing] randomized ones (which are worthless for tracking). Other cookies that just contain your search preferences will be allowed, so your preferences are preserved." The developer also notes that the NSA can track you by your IP address or browser headers, but these techniques "are not as accurate as tracking IDs that are sent on every search."
Searchonymous does not install a button to Firefox. To turn it off, you'll need to head to Tools > Add-ons and disable it.