What is the easiest, most effective backup system for Ubuntu Servers?

The best backup solution is highly dependent on your requirements. It's best to start with a backup strategy and let your strategy inform your choice of software. At minimum, you'll want to read enough to know the difference between the common backup types (incremental, differential, full).

If incremental backups aren't important, rysnc + mysqldump should work fine for your scenario. If you really need/want incrementals, Amanda or Bacula are good choices. My company has used Amanda for its backup needs for several years, and we've been very satisfied. Zmanda also provides a tool for backing up MySQL databases.

Whatever solution you do chose, though, test your backups! Don't just check to see if the backup succeeded, do a restoration (preferably from "bare metal") to verify the files are restored correctly. There's nothing worse than discovering your backup is incomplete when you've just clobbered critical files.

Also, if you're backing up a production environment, make sure at least one of your backups is an encrypted off-site backup.

Good luck!

In my opinion backuppc is the best backup solution for multiple server.

Backuppc uses a dedicated server to backup different hosts. Every hosts can be configured separately (what to backup, backup schedule, transfer method (including rsync), ...)

pre and post backup jobs can automatically be launched (e.g. dump mysql tables)

administration is done by web interface

installation is done by apt-get install backuppc

Try rsync, I think it will provide all the functionality you're looking for. In order to handle MySQL you might need to write an additional cronjob to dump the database as well.