Being able to solder is a really useful skill, if only to be able to make up simple audio cables and replace broken connectors.
As for your linked video, I'm not too sure about that guy's wire-stripping technique, nor why he's using such thick wire! You might be better looking for some "how to solder electronics stuff" guides online.
You don't need a fancy temperature controlled iron for this kind of thing, just something with a fairly small tip (say 1/8" or less). Use thin solder with a flux core that's designed for electronics, not plumbing. The trick is to coordinate applying the iron, then just enough solder, and then quickly removing the iron. Just practice on scraps of hook-up wire and offcut component leads. BTW, unlike in the video, most of the time you don't need to wrap the component leads any more than is necessary to get them to stay in place while you solder, as you don't need great mechanical strength, just a good connection.
For basic audio use, e.g. making up an SB patch cable as discussed, ordinary 1/4W metal film resistors are fine. You just need components with wire leads, and not the surface-mount variety (same with capacitors). It's a pity we don't have local electronics stores anymore, or even any back-street ham radio suppliers, as they were great for this kind of stuff. But if you do know anyone locally who's into electronics, that would be a good start. Sorry I can't be of any practical help, as I'm the other side of the pond!
Good luck with your project!
Martin