I would recommend after getting the Back up and Migrate SQL dump from your current site,
Go to your new host and upload all your files,
Go into sites->default and rename the settings.php file to something like old-settings.php
Then make a copy of default.settings.php and rename it to settings.php
On your new server, create your DB and DB-user
Then run the Drupal install like you were installing a new site
Then import the SQL dump file from the old site
That should do it.