So, took one of the stations, some new 18g wire and some new solder. Cut about 10 3" strips of wire. Stripped both ends of each wire and soldered the ends together. The first one was a mess, the second one a little better, and by the end I started to get good looking, shiny joins.
I got better results when I stripped more of the wire and made a clean, good looking and strong physical connection first. I had to strip more insulation on either side than anticipated to do this. I guess next step is more practice and some heat shrink tubing. Or would electrical tape be just as effective? I'm not going to use these practice strips for anything but I believe in practicing the way you play.
I still haven't gotten the hang of many of the techniques described above, though. The only consistently effective way I found was to make the solid connection, wrap a bit of solder around the connection, and then actually touch the iron to the solder to melt it into the wires. No combination like solder on top/iron on bottom would work. The iron was on 40w.