Here’s an overview of how Linux reads its filesystems on boot:

1. You press the power button on your PC.
2. Your motherboard receives power and begins a process called Power-On Self-Test, or POST. This is essentially the motherboard “feeling out” on all of its ports to see what’s connected. It senses all of your hard drive(s).
3. The motherboard picks one of the partitions on one of the connected drives and tries to run it as an operating system. For a linux system, you want this to be /boot.
4. /boot finds your actual system partition and tries to spin up the OS using it. This partition becomes the “main” one you’ll see by default when it’s fully up and running.
5. At some point in the boot process, Linux checks the contents of the file at /etc/fstab. “fstab” here is short for “file system table”. It’s basically a list of other partitions plugged into your PC, and a mapping of where it should be mounted in your filesystem. If your system partition was Partition A, for example, and you had another Partition B that you want to be accessed at the path /B, you’d add a line to your fstab file that says something to the effect of, “hey, reach out to the connected partition with ID <long id string>, and if it’s there, create the folder /B, and make the files inside of the partition show up there.” This is called “mounting” the filesystem. You can do it manually at any time with the mount command. The fstab file is just a way to get Linux to auto-mount permanent partitions on startup. You can mount any drive to any path, including /home.

What you need to do is:

1. Format your new drive, and create a new partition there.
2. Mount the new partition with the mount command. Park it anywhere you want. /media/new_home or whatever you like. This will just be a temporary place.
3. Copy everything from your /home directory to the newly mounted one.
4. Move the existing /home folder to literally anyplace else than where it currently is. \home_old would do. It just needs to be out of the way. You almost certainly want to do this only when logged in as the root user, or from another OS running off a USB. You will not be able to log in as any user after you do this until you finish the following steps.
5. Edit your /etc/fstab file to point your new home partition to the path /home. This will “hook up” the new partition.
6. Reboot to make Linux mount the new partition.
7. Verify everything works.
8. Delete the backup home directory.

Answering some of your questions:

Can /home live on a separate drive from the system partition?

Yes. Linux does not care in the slightest where any mounted drives are. A drive is a drive is a drive. If it mounts, it mounts. Just make sure it’s in your fstab file.

How should I organize my partitions?

There is no right answer. It depends entirely on what you need and how you intend to use your PC. But since that answer is unhelpful, I’ll tell you how I’ve done mine. I currently have a single 2TiB SSD split up like this:

[ 200 MB boot | ~500 GB system | 2 GB free | ~1.5 TiB home | 8 GB swap ]

Boot comes first and is tiny. System is probably an order of magnitude bigger than it ought to be, but whatever. 2 GB of deliberately unallocated space is there for the exceedingly rare situation where a dumb bug chews up all the storage on the system partition rendering it un-runnable; it’s emergency expansion space I can tack on to get it running just long enough to resolve the problem. Home gets the lion’s share. And swap goes at the end, where home can cannibalize it if some day I decide I don’t need so much swap space.

You would probably organize your drives in a similar way, except one drive would be entirely dedicated to /home and nothing else, and system taking up the bulk of the other drive. That would be the easy and naive solution. Some users may be inclined to create other partitions for organizational purposes. I personally don’t care for that organizational pattern, so, w/e.

I cannot help you with your display troubles. :(