need to test if sql query was successful

This is the simplest way you could test

$query = $DB->query("UPDATE exp_members SET group_id = '$group_id' WHERE member_id = '$member_id'");

if($query) // will return true if succefull else it will return false
{
// code here
}

mysql_affected_rows

I understand that by saying "successful" you want to know if any rows have been updated. You can check it with this function.

If you use PDO -> http://www.php.net/manual/en/pdostatement.rowcount.php

global $DB;
$status = $DB->query("UPDATE exp_members SET group_id = '$group_id' WHERE member_id = '$member_id'");

if($status == false)
{ 
    die("Didn't Update"); 
}

If you are using mysql_query in the backend (whatever $DB->query() uses to query the database), it will return a TRUE or FALSE for INSERT, UPDATE, and DELETE (and a few others), commands, based on their status.