Cookies and IP addresses may not give away your personal information when browsing, but when these clues are combined with other data, like your search history, you could unwittingly disclose your identity to hackers, scam artists, or investigators. Search engines routinely store search queries associated with your IP address. Aside from using a proxy server to replace your IP address with its own, there are some other ways to be online anonymously.
Remove what is stored
When you surf the Internet, a lot of what you do will be stored on your computer. Someone who uses your computer would be able to find out which sites you visited and possibly even user names and passwords you typed into websites. One of the ways to cover your online tracks is to manually delete your browser history, which all browsers allow you to do. You can even set up your browser to do this for you automatically (at the end of each browsing session after you've quit the application or logged out of the machine).
Another option is to use anonymous surfing to prevent leaving any kind of trace at all. Once enabled, your browser won't save your surfing history, search queries, cookies, download history, or passwords. Depending on which browser you use, this feature will be called either "private browsing" (Safari and Firefox), "InPrivate browsing" (Internet Explorer), or "incognito browsing" (Chrome).