20 AWG is the largest I've ever heard anybody make headphone cables from, and that was for an HE-6, which is already so large and heavy that the added mass of the cable isn't as significant. I don't think there are any phone plug housings that could fit three strands of 14 AWG...
You don't want to use anything larger than 24 AWG.
It is fairly easy to get small spools of hookup wire (
for example), but much hookup wire is solid-core, which is bad for headphone cables.
I've used "speaker wire" from hardware stores before, and keep some handy in a scrapbox (there's some working as a grounding strap from my receiver to the wall as I write this). It's usually 24 AWG or so -- it's actually just zipcord with a clear rubber insulator and one of the lines is tinned -- and does not look or feel nice at all -- but it is cheap, and might be something good to practice your soldering with.