Are there any tricks to transferring and/or sharing the Resource Links across multiple PHP pages? I do not want the server to continue connecting to the database just to obtain another link to the same user in the same database.
Remember that the link returned from mysql_connect() is automatically closed after the script it originated on completes executing. Is there anyway to close it manually at the end of each session instead?
PHP allows persistent mysql connections, but there are drawbacks. Most importantly, idle apache children end up sitting around, holding idle database connections open. Database connections take up a decent amount of memory, so you really only want them open when they're actually being used.
If your user opens one page every minute, it's far better to have the database connection closed for the 59 seconds out of every minute you're not using it, and re-open it when needed, than to hold it open continually.
Instead, you should probably look into connection pooling.
One of the advantages of Mysql over other heavier-weight database servers is that connections are cheap and quick to setup.
If you are concerned about large numbers of connections to retrieve the save information, you may like to look at such things as caching of information instead, or as well as getting the information from disk. As usual, profiling of the number, and type of calls to SQL that are being made, will tell you are great deal more than anyone here guessing at what you should really be doing next.