Here are the steps

 

1. Zip the content on your old site. Eg abc.zip 
2. Download this attachement on your PC.
3. Open the config file of your website ( Different for different CMS ) 
4. Note the DB Name, it will be like black_wp1, etc
5. Go to PHPmyAdmin, and open this DB. Click on extract. Extract it to your PC. Eg black_wp1.sql

6. Now go to new hosting
7. Create an addon Domain
8. Upload the content to the file manager
9. Click in Databae Wizard. Create a new DB and User and Link them with all previliges. In cpanel u will find 2nd option Database Wizard. Just proceed with it it will create.
10. Open PHPmyAdmin on new hosting 
11. U will find the New DB u just created. 
12. Open it.
13. CLick export and select the DB u just created and click GO.

14. Go to file manager, open config file ( Different for different CMS ) and change the new db name, db username, password.

15. Change NS.