What's the best way to store temp data in PHP across page views? MySQL or server side cookies? Or something else I don't know about?
You could use the $_SESSION variable. I've seen people using it to keep session information such as cart contents, and generally to pass information from one page to another.
There's an extensive documentation on session variables at http://www.php.net/manual/en/ref.session.php .
I would advise against MySql in this case.
I prefer using PHP session for storing data on server side. For efficiency you can use memcached to save session values (default are saved on filesystem).
You can use $_SESSION like Clement suggested, but $_COOKIE may be appropriate also. Especially if you need to fetch the values client side. Here are some examples on how to use Cookies in PHP: http://www.w3schools.com/php/php_cookies.asp
Session variable is backed by a storage mechanism, that is, when the request finishes the session gets written by the session handler, by default this is to a file. On the next request it is pulled back from that file (or whatever else the session handler uses).
If you're reading and writing this data on every request, just stick with a the $_SESSION variables, the overhead of connecting, querying and updating a database will not be faster than the default $_SESSION.
You'll probably only ever want to use a database backed session if you are running multiple load-balanced servers and need to share the session data between them. In this case, if you find the overhead of the database sessions to be slowing down your site to a noticeable degree you might consider sticking memcached between your web server and the database.