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Hi,
Try the following:
Check your PHP / MySQL version on the old server (save phpinfo())
Check your PHP / MySQL version on the new server (save phpinfo())
Use FTP to get all your files to your local PC
Use PHPMyAdmin / MySQLDump / BigDump to dump your DataBase and download it to your local PC
Upload the FTP files to your new server
Restore the dump on your new server
I would use the local PC as test setup to see if everything is still working. Maybe PHP version on new server is PHP5 and on old server it is PHP4, then you might get into trouble. Same goes for MySQL.
If you use local PC as test setup, use same versions as new server and see if it works.