Here are some links in regards to backing up the database/restoring the database:
From the manual:
Backing Up The Database via SSH/Telnet or
Backing Up The Database via phpMyAdmin
Restoring The Database via SSH/Telnet or
Restoring The Database via phpMyAdmin
If you don't have shell access, some people have also reported success with these scripts:
MySQLDumper
MySQLHotxcopy
Bigdump
Don't forget that if he has attachments or avatars, etc, stored in the filesystem, then you need to backup those also.
And read this in the manual -
Moving Servers