So I pulled out the Solder Practice Kit that I had purchased a long time ago with the best of intentions, and decided to try it out. Several hours later, I had blinking lights. It definitely took a while to get the hang of, but it wasn't as difficult as I thought it would be, and it was fun! For someone who has spent a long time in software, it was incredibly satisfying to build something physical. I feel like a new world was opened up for me, and I hope to do some more substantial projects in the future.
Here's the proof that I actually got it working: