Just like the guide says, this is the most successful path to upgrading modern Fedora versions:

sudo yum update fedup fedora-release
sudo fedup --network 20

This upgrades from Fedora 19 to 20. Adding the --reboot option to fedup will automatically restart the system. You should then see a System Upgrade entry in grub’s menu. Selecting it should continue the installation process, replacing your previous packages with the upgraded ones. I can personally vouch for a smooth experience.

I dual boot Fedora and Windows 7, and apparently Grub 2 has been having issues on Fedora that prevent you from editing the boot menu, so previous kernels stay listed. Has anyone else dual booting with Fedora found a way to deal with this?