I have three MySQL databases and I wish to connect to all three sometimes at the same time. How do I open three separate connections to the databases?
It depends on your database API. For example, with the PDO extension you would create a PDO object for each database connection. Likewise, with the MySQLi extension you would create a MySQLi object for each database connection. With the legacy MySQL extension you would have a different link identifier for each database connection.
I have noticed that some threads mention ADODB. Is this faster than the traditional method in connecting to a MySQL database?
No, it is an abstraction layer and thus may even result in slightly lower performance as a trade off.