Cookies are basically a text file stored on your computer. The data they contain is sent to webservers that the cookie says it should be sent to. The way browsers and cookies work, only the cookies for a certain website will be sent to that website. If someone used some sort of cookie stealer to steal cookies, those cookies would be for the domain in question. So, in short, just visiting a page, unless the browser itself is vulnerable (like Internet Explorer), will not reveal cookie data and browsing history. For a site where your cookies were stolen, yes, changing the password for the account will usually prevent the cookies from being used to acess your account.
It's not your resonsibility to protect yourself from cookie stealers as long as you use up-to-date browsers, such as Firefox. If your cookies are stolen, it's because the site or server itself is insecure.