Back up your directory and when you move it over to the other server keep the file structure the same from root up. This way you don't have to go through the SQL files to update file locations / preplan it. You redirect the domain to the new nameservers and within 1 hour - 24 hours that domain should be redirected at the new host.
Back up your SQL database with something like phpmyadmin and just do a database dump and make a new database on the new server and upload your database there. Goto your wp-config file and make sure your database connection information matches with your new host.
Make sure you do most of this before you switch the domain name if you don't know your servers ip if it uses cpanel, or others etc..
You can also move over all the files and the database and connect everything before the server and new domain connects. So when it does swap there won't be any downtime.
After that you should be good to go. Hope that helps.