Firstly, if you haven't already, make sure that you're not using LVM, as this complicates things.
should return something like "/dev/sdxn
on / type ...". If it says "/dev/mapper/..." then things are going to be more complicated.
Assuming that LVM is not being used, the easiest method is to boot the live CD and use GParted to change the partitions. If the live disc doesn't have GParted, install it in a terminal:
su -c "yum install -y gparted"
Use GParted to shrink your existing root partition as much as you can, then create a new partition for /home (type ext4).
Then you need to mount both partitions. GParted will tell you what the partitions are called, so adjust these commands as appropriate, but I'll assume that the old partition is /dev/sda1 and the newly created one is /dev/sda2.
mkdir home root
mount /dev/sda1 root
mount /dev/sda2 home
Move the files in /home:
mv root/home/* home
Force an SELinux relabel on the next boot:
Get the UUID of the new home partition:
To make sure that the new home is mounted on boot, edit /mnt/root/etc/fstab and add the line:
UUID=uuid_from_previous_command /home ext4 defaults 1 2
Then unmount the partitions again.
If you want, you can then resize again in GParted to shrink / and grow /home now that the files are no longer in the root partition.