If you connect to a database with each script, consider using a persistent connection rather than a normal connection. For MySQL users, that is the difference between using mysql_pconnect() rather than mysql_connect(). Persistent connections remain connected even after your script has ended, which means that the next time a script asks for a connection, it uses the one that is already open - this saves a lot time negotiating passwords and such that can otherwise be used to execute important code.
Switching to persistent connections does not require any other change than adding a "p" in the function name - the parameters are still the same. If your database server is not on the same machine as your web server, consider using CLIENT_COMPRESS as the fifth parameter to your mysql_connect() /mysql_pconnect() call - it allows MySQL to compress data to save space, can drastically lower network bandwidth and transfer speed, particularly when reading in lots of data.