I am currently being hosted by a company that uses your same control panel. How can I transfer all of my files including my ftp/email/site files to here?

CPanel can automatically transfer domains from other server with CPanel contorl panel. But to do this I need to know the root password for that server. I doubt that sysadmin of your current hosting company wants to give me this password : ))

So the only way is to upload all files through the ftp and do all configuration trough control panel.

Ok, thanks.

Can I use the Cpanel feature to download today's home backup? If I do that will it backup all of my files including email, http files, etc...

Yes, you can download backup of your home directory (including mail and http) and backup of your database from your control panel.