Moving prestashop to another host is an easy process but can also make your head spinning especially if your site is using Smarty templates.
To move the site and make it work you only have to follow the following steps
1) Move all the files (use FTP or any other method)
2) Export and import the database to the new server.
3) Navigate to site/config/settings.inc.php and update the database details. Remember to update the __PS_BASE_URI__ as well to reflect your current settings.
4) If you are using Smarty templates, MAKE SURE you enable write permissions for the entire folder.
Be careful with permissions step if you are going live