Step 1: Get the files backed up from your ftp onto your desktop in a folder, then back up your database for the wordpress. (Save it as a sql.gz)
Step 2: Get your new host up and running and load the backed up files onto your new server.
Step 3: Create the database of the wordpress and load the backed up one you have saved.
Step 4: Load the database on your new server and make sure that the config settings for the wordpress matches the new database name from your host. Ex: username_databasename
Step 5: Point your new domain to the new hosting and make sure it is in there correctly, wait 12-24 hours to allow the DNS to kick in.
Step 6: Leave the files on your existing host and then when your new one has kicked on, you're in business.
To check if your DNS has been completed, use
http://www.whois.com
Hope this helps!