The plugin I use to backup and transfer wordpress is called XCloner. It is a free plugin that will package all wordpress files and its database.

1. Install XCloner on wordpress
2. Add directory in root administrator/backups
3. Setup XCloner and change default password
4. Backup Website with Xcloner
5. Move backup file to new server
6. Create a new mysql database
7. Copy restore file to the new server ( you can find there in the xcloner plugin folder)
8. Go to the new server address /XCloner.php to run the setup wizard
9. Verify that wp-config.php has all of the correct information for the new server
10. Verify that wp-options table has the correct website address information.