1) Put your site in maintenance mode if you want to make sure nothing new happens while moving, as this would be lost (user signing up, creating content etc)
2) Dump your database. (mysqldump -u username -p DRUPAL-DB > drupaldatabasecopy.sql)
3) Copy the filestructure in the drupal root from server A to server B
4) Create a database with the same name on server B, and make sure it can be accessed with the same credentials as on server A. The credentials can be found in your settings.php
5) I suggest using a hosts entry to verify things are working (
http://en.wikipedia.org/wiki/Hosts_file)
6) All is well? Change your DNS-pointer :)
These steps may differ depending on your environment, but mostly, it's not hard to move Drupal around :)
