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!