Plugin with connection to database - Single function

Use a global.

So, your plugin file would look like this:

$myConn = new wpdb( 'username', 'password', 'database', 'localhost' );

function plugin_step_1( $arg1, $arg2 ) {
    global $myConn;
    //code to do stuff here
}

function plugin_step_2() {
    global $myConn;
    // more code here
}

function plugin_step_3( $arg1 ) {
    //I don't need the wpdb object
}

add_action( 'init', 'plugin_step_1' );
add_action( 'template_redirect', 'plugin_step_2' );
add_action( 'wp_head', 'plugin_step_3' );

Of course, your functions wouldn't necessarily need arguments or whatnot.

Also, note that the way MySQL works, connection persistence may or may not take place, even with a global wpdb object specified in your plugin / functions.php.

Connection persistence (that is, keeping the connection open until all queries are completed for a single page request) generally takes place during an entire page request, but based on when and how your plugin / functions.php goes about calling event handlers for a given request, it may be that WordPress effectively closes the connection before all requests have been made, and thus will open more than one connection, even though you are using a global object to connect.

But, it sounds like your need is mostly about keeping things DRY, rather that persistence or pooling, so this should do the trick.