18AWG is not particularly robust wire, you should have no problem heating it enough for this to work with pretty much any cheap-o soldering iron.
It's possible that the wires are contaminated, but dipping them in flux won't have much more effect than the 60/40 you're using, which already has a rosin flux core.
I tend to only use flux when soldering copper pipe, as it really is useful when soldering with a torch. In my experience a torch--even the little creme-brulee making ones--are a little too hard to control and I just wind up melting the insulation and burning my fingers.
I use a solid-core silver solder myself for all the electrics that I am working with, and I've noticed that it adheres a little better. I find the flux core is prone to sort of envelop the little bead of solder and make it want to slide off of my work; that could be what's happening.
As well, as you're new to this, don't forget to slip the shrink wrap sleeve over one of the ends of wire BEFORE you do your soldering, or you'll be doing it all over again. It's the most common mistake I see when I'm teaching new students.
Agreed on all points.
I've been lucky enough to inherit some old school lead solder from my father (RIP). That stuff works the treat.