Currently the elbows are pointing in vaguely the right direction but they are somewhat off. I have checked that they have the right target bone selected. I have a feeling that this is due to some of the rotations of the bones in edit mode... Can someone help me with this?
You'll need to adjust the roll of the bones in edit mode. Ctrl+R let's you adjust the roll, and if you have a more complex rig based around parenting thd deform to more advanced control and mechansim bones like ones from the Humane Riggin course, youll want to adjust everything in the mechanism. You'll probably have to go back and forth a few times between edit and pose mode.
Also definitely check out Rigify, it's not some proprietary addon, it's Blender's own rig built into the software. It's really powerful but it's also kind of a lot all at once. There's a lot of resources and tutorials out there that go pretty deep into how rigify works and how all of the setup components can interact. If you van learn how to build out the metarig before generating the full controlled rig, you'll be able to rig damn near anything.