Ok, there's all kinds of fancy devices available for rebuilding, and I have most of them. They make it more convenient, but aren't truly necessary.
My first build kit was a set of precision screwdrivers, toenail clippers, scissors, a roll of wire, a cigarette lighter, and a bag of cotton balls. When it comes right down to it, the cigarette lighter wasn't really necessary, just nice to have.
Cut a piece of wire a few inches long. Generally, for a single coil, one wrap from a spool is a good length. Grab an appropriately sized round object (such as a small screwdriver), hold the object and one end of the wire, and then wrap the free end of the wire around said object with the other hand. At a minimum, that's all you need.
Having a build base can be handy, but I rarely use mine. I put the atty on my mod and build there. I hardly ever touch my ceramic tweezers, mostly because I rarely make contact coils. When I do, I just heat the coils with a lighter until they glow, and then install them, which is usually enough to get rid of most hot spots.
However, I have switched to using a coil jig, because it makes it easier to get nice, even wraps. To space the coil, I use my pocketknife to gently pry each wrap apart, until they're spaced to my liking.
Out of my whole kit, with all the fixings, I use the coil jig, a pair of cutters, a pocketknife, an Allen key / screwdriver for installing the coils, and scissors to cut wick.
BOOM!