How can I move my /home directory to another partition if it's already part of the / partition?

I think it's best to follow the Ubuntu wiki guide. I used it a while ago and it worked just fine.

Yes, it is only a matter of adding a new entry to /etc/fstab and then copying the files over.

I suggest using UUIDs for the partition identifier in fstab, the syntax similar to this:

UUID=abcdabcd-acbd-abcd-abcd-abcdabcd /home ext4 defaults 0 2

Copying is best done as root, with "-a" flag passed to cp. Also, better play safe and not remove the files immediately:

cp -a /home/* /path/to/new/partition/
mv /home /old_home
mkdir /home

Note that the user directories must be straight inside the partition, not in /partition/home/