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.