1) Copy over the files. You can use FTP for this. Or you can use shell access if you are comfortable with that.
2) Copy over the database. This requires manually backing up the database on the old server, copying over the backup file to the new server, and then restoring the backup file on the new server. I recommend using shell access to run the backup / restore commands if your host allows you shell access.
Sorry, but here's another bump of this old thread...
I just moved my install to another server doing it exactly how it was described in this thread. I updated the info in the config.php file with the new dB name and password, but I cannot get into my Admin CP or board in general. When I try to access my site it says "An unexpected database error occurred. Please try again later."
It's not the server and I have uploaded all of the files and directory structure to the root and have installed the SQL dB backup file correctly using the restore function in my new cPanel. I did have to rename it though because it renames it anyway with the new prefix and I created a new dB user with new password. Again, I've edited the config.php file with the updated info.
How can I get into my Admin CP or rebuild the cache?
Turns out that when I transferred everything to the new server I also transfered the php.ini file from the old server as well so it was messing up the communication to the extensions on the new server.
Was just a brain-fart on my part.
Lesson to be learned, don’t transfer your php.ini file .