Download the all-in-one WP migration plugin. It will help you move the site easily enough. When doing the backup you can do a search and replace to catch your URL string and update it with the new one, then install WP on the new host and the plugin and import again. It seems pretty robust. I used to do it all manually, zipping files, exporting SQL, editing etc, but it's long winded.
edit: beaten like a ginger step-child