The best safety advice for using public computers or public WiFi is not to use any public infrastructure when your password is involved. Hotel computers, terminals in airports or at internet cafes cannot be secured. The computers of friends and relatives might be compromised. Using your own computer over someone else's internet connection also holds risk. If you must use either public terminals or public WiFi, there are things you can do to mitigate the risks.
Use Private Browsing, secure connections, and carefully manage caches and cookies in order to minimize risks when using untrusted computers or public internet.
Private Browsing masks your browsing trail from anyone who might later use that browser and try to check the browser's history, cookies, and other data to compromise your security and privacy. Using Private Browser and closing the browser when you are through also logs you out of web based applications you had been using. Shutting down the browser is insufficient, unless you are in Private Browsing mode. If you use Private Browsing mode, you don't need to worry about clearing your cookies and caches. Enable Private Browsing before beginning browsing. Browser specific instructions for using private browsing.
Make sure your browser connection is a secure one. If you cannot connect to the website with https:// (note the "s" = secure) your browser connection is not secure and the traffic going to and from your browser can be easily captured and read. Many websites, such as google, use https:// by default and any time you deal with sensitive information like personal records or banking information the site should be secure.
Emptying caches and removing cookies is an important step when using a public browser. Stored cookies and your web history can be used to compromise your security and privacy. If you cannot use Private Browsing functionality, the following should be done after you finish browsing. Browser specific instructions for managing cookies and caches.