The lower the gauge number, the thicker the wire. 20-22 gauge is normally for power wiring, and 26-30 are for general wiring. 26 is standard IDE wire, and 30 is kynar or "wrapping" wire.
solid core is more rugged, holds its shape, but can brake if flexed to much, stranded can be flexed around easily without breaking. it mostly depends on the aplication you need it for
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You're probably not going to want shielded wiring. That has an outer "wrap" layer of wiring to reduce electromagnetic interference, but you won't need that for short portable distances and it will just get in the way.
Most portabilizers I've seen use old IDE wire- the kind that connects old floppy and hard drives in computers. These guys:
Snip off the ports and you're left with thin, separable lines of wire. Do you need to run five wires between two places? Peel off a five-wire-wide strip of your IDE, strip the ends (you can usually do this with sharp finger nails because the plastic is not durable), and get to solderin'.
If you don't have any IDE, get a high-gauge spool of what's called hookup wire- 22 or higher, preferably stranded.
Get the 26 gauge. You don't need a whole lot of 22gauge because there's only a few power lines you need to wire. Sheilded is really only needed for video lines to reduce interference.