Log in as root to avoid trying to make sure no files in /home/ are being read/written to.
Step 1: copy data to new drive.
Mount new drive to /mnt/.
cp -ra /home/* /mnt/.
-a means that all permissions remain the same which will mean that your user can still read them. Check the man page for more details.
This command will take a while. Use -v to see progress.
You should see a folder with your username appear.
Step 2. Prepare /home for new drive.
Move the files to a new folder. This is done to make sure you can still easily go back.
mv /home/ /home-bak/ keep your old home dir safe in case a mistake was made.
mkdir /home/.
Step 3. Mount your new drive.
Mount your drive to /home/ and check if you can login.
If everything went correctly, you should be able to just login.
Finally you need to update your /etc/fstab to include this new drive. This will make sure your home drive mounts when you start your os.
If everything is working, you can delete your home-bak as well.
the mv home is just renaming the folder so you can mount home in the same space. Rsync is probably better than cp but I didn't want to suggest tools that op doesn't have installed.