This should do it
Step One - Backup current database:
Telnet into your system
from the root directory, type mysqldump -uUSERNAME -pPASSWORD databasename > /path/to/dump.sql
this should only take about than 30 seconds to a few minutes, depending on your database size
when it is completed, it will bring you back to the prompt
verify that you have a dump.sql in the /path/to area you specified
Step Two - Transfer to new server
telnet into machine1, and from the command prompt type the following:
machine1$ cd /path/to/yourbackupdirectory
machine1$ ftp machine2 (IP or host) - this should open a connection and ask for your username/password
ftp> bin
ftp> cd /path/to/newdirectory
ftp> put dump.sql (should take no more than 30 seconds to a few minutes, depending in the file size)
ftp> close
ftp> quit
verify dump.sql is in the /path/to/new area on the new server
Step Four - Restore to new server
telnet into machine
create new database on the mysql (either via phpMyAdmin or telnet)
from the root, type mysql -uUSERNAME -pPASSWORD newdbname < /path/to/dump.sql
this takes a couple of minutes depending on your database size, but you can track the progress either by FTP (checking the /mysql/data/newdbname and see the files appearing) or by phpMyAdmin (you should see the files appearing there)
when it is completed, it will bring you back to the prompt