Hello clmirand,
If you are using phpMyAdmin for both of your mySQL databases, you an export the database from one of the phpMyAdmin pages and download it directly to your computer. You can do this by going into phpMyAdmin and selecting the database you are going to export. The export button is on the top row of tabs, about half way over, called 'Export'. You shouldn't need to change any settings.
In the new account, create a database named the same as the one you have exported. You can then go into phpMyAdmin, select the database, and click on 'Import', also located on the top row of links in phpMyAdmin. You can then upload your .sql file that was downloaded from the other phpMyAdmin. Once you have imported the .sql file you should see all the tables appear in your database.
Let me know if you have any problems, or are not using phpMyAdmin.
Jake