This method just uses the whole mysql database rather than a WordPress export/import from within, and is best suited for a straight swap. So you would copy all the WordPress files/folders to the new destination, set the correct ownership to those files = then do the database switcheroo.

Do a mysql database export of the old database on the old server, create a new blank database on the new server, import the old data either in phpmyadmin or mysql directly in the command line.

Make sure you have the new database selected, then run some sql updates and replacement commands on the tables notably, wp_options, wp_posts, wp_postmeta.

Use the code as below and swap in your old and new URLs, no trailing slashes. Also if necessary change the table prefix values where applicable (ie wp_ )

UPDATE wp_options SET option_value = replace(option_value, 'http://www.oldurl', 'http://www.newurl') WHERE option_name = 'home' OR option_name = 'siteurl';

UPDATE wp_posts SET guid = replace(guid, 'http://www.oldurl','http://www.newurl');

UPDATE wp_posts SET post_content = replace(post_content, 'http://www.oldurl', 'http://www.newurl');

UPDATE wp_postmeta SET meta_value = replace(meta_value,'http://www.oldurl','http://www.newurl');