Moving OpenCart installation from one server to another may seem difficult at first, but it's actually pretty straightforward.
One possible way of doing it is described in OpenCart documentation, however, we're using a different approach.
First make a dump of your OpenCart database using mysqldump or any other relevant software.
Then copy the whole OpenCart folder including database dump to your new server.
Modify the files config.php and admin/config.php on your new server. You'll have to change URLs in the first five configuration variables (HTTP_* and HTTPS_*), paths in the next ten lines (DIR_*) and database credentials in the last five entries (DB_*). Simple search/replace will do.
When you're finished with that, import your database dump into your new database.
Basically that's it for non-SSL installations. If your server is configured correctly, you should now be able access your new store.