Which of the UX courses should I take? Which one should I take first?

Of the two 4-week UX courses we currently have, the Interaction Design course is more focused on web/mobile work (we cover concepts like usability, wireframing, sketching, information architecture, site maps, user-centered design... which are all mostly in the context of creating websites or mobile applications).

The Research course is pretty broad in its applicability, in that you'll basically learn the design thinking methodology in depth and cover, end-to-end, the process around exploring a problem space, understanding customer needs with various techniques, and brainstorming / testing solutions to those problems that you've discovered. You can use that framework and those techniques in basically any context, whether or not you end up creating an app or website at the end.

For even more info, looking at some past student work in both the courses might be helpful:

Which course you take likely depends on your UX goals and what you're looking to add to your skillset. The IxD course would be a good fit if you're interested in doing things like creating wireframes, understanding how to create cohesive site flows, etc. You can take the courses in any order.