Do you only have to tell the new host to add the site and they will take care of everything as far as dealing with the current host.
Upload every thing to da new host, update da DNS records, then when every one is accessing da site on da new server, dump da old host!
So, I basically need to upload all the files that are currently on the existing server to the new server and then update the DNS records, right?
Yes, but if you are using MySQL databases, export database from old host using phpmyadmin & then import it to the new host again using phpmyadmin. Any config files will need to be updated if you have changed the username (of FTP/cPanel) or the database names & passwords.
^^ True. Your best bet is to generate a full home directory back up from your cpanel (assuming your using that) and download it to your pc, and as Ishan advised, go to phpmyadmin and export your database and create it again at your new host. Once everything is all set, change your DNS nameservers.
For a full backup, you don't have to separately transfer the database. Download the full backup. Send the downloaded file to your new host and ask them to restore the backup. Most of the hosts will do the whole transfer themself, if you ask them. After restoring, change your DNS records and your site will be up again once the DNS updates.But till the DNS updation takes place, which may take a few hours at times, make the site will be available from the old host. And any new entries added to the database will be unavailable once the updation takes place. So make sure that doesn't happen. And as iShan said, you will have to change the document root, dbusername and dbpassword if they are different from the current ones. You don't have to worry about the dbuser and dbpass, if you use the backup restoration method. But if you have have used absolute path somewhere in your script, then that will create problems. In cPanel to cPanel transfer, this can be avoided by signing up with the same username, you have with your current host.
This sounds a bit complex. I thought about uploading all the files to the new server from the hard drive of the computer where all the files are also stored? would this work? If this does work, do I upload all the files or just the all the content files?