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On the old server, In phpMyAdmin, go to the section where it says "View dump (schema) of database" (You need to select a database on the left)
Click the "Structure and data" radio button and check the "Send" box and click Go. It will make your browser download a .sql file.
On the new server, go into phpMyAdmin and select a database on the left. Then, go to the section that says "Run SQL query/queries on database [database name]" and click the Browse button. Select the .sql file that was saved and click the Go button. All the data from the old DB will be imported into the new DB