I was able to use TechShop's soldering stations and helping hands to keep everything in place while I worked.
Step 1: Prepare your wires
First, the filaments of each wire should be twisted together, or pigtailed, to keep them orderly and behave more like a single entity.
I prefer to add some soldering resin to the filaments at this point too. It really helps the solder flow when tinning the wires. If using flux core solder, this step isn't critical, but it still does help.