I'm not sure how this is supposed to be done. Right now I'm just doing a

$db = mysql_connect( $dbServer, $dbUsername, $dbPassword );
mysql_select_db( $dbName, $db ) );

at the top of every page that accessed the database.

I've read somewhere that the database connections add a lot of overhead. Is that true? I've also read about establishing persistent connections with pconnect(), but are there any downsides to using them? It almost sounds to good to be true.