On the new host
It is probably a good time to do upgrade where you did not in the past. It is not needed to install the exact same version of Drupal in your old host. You can install new version of Drupal directly as long as they are compatible.
Before move on, please make sure the new website for Drupal is configured correctly.
Install latest compatible Drupal Core version
This can fix many security issues and more stable. Follow the official Drupal documentation to finish the procedure. Note in case of Drupal 7, you need to manually create database and user account from mysql command line before let Drupal config your database.
CREATE DATABASE databasename;
CREATE USER 'username'@'localhost' IDENTIFIED BY 'password';
GRANT SELECT, INSERT, UPDATE, DELETE, CREATE, DROP, INDEX, ALTER, CREATE TEMPORARY TABLES, LOCK TABLES on database_name.* TO 'username'@'localhost';
Make sure the database name and user account is the same as in your old host.
Verify that the site now works by input the domain name of Drupal site in browser. Only basic view and modules are presented so far.
It is time to import the database from old host now. Log into mysql and import it using command line:
Install extended modules
Install all extended modules as in your old host. Use latest compatible version of the module if available.
Enable all necessary modules.
Copy all files under the file directory in the compressed package from your old host:
to the corresponding path in your new site.
Copy all files in the compressed themes package from your old host into the corresponding path in your new site.
OK, now try to launch the new site and see if everything works fine!