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.
Step 2: Tin your wires
You want even distribution of the solder between the filaments, but not so much that the solder starts to swallow the filaments. The top wire in the picture is just about right, but the bottom one has too much solder. Too much solder will make the wires quite stiff, and they still need to be fairly pliable for the next step.