I use my gas stove for heating coils and oxidizing SS wick.
If you have a small set of drill bits winding coils around them is fine; in a pinch you could use a toothpick or any small rod like material.
If you want to have a nice set of mandrels, most hobby shops sell wire rods in an assortment of diameters that you might like to use. Another option is tool stores that sell drift pin punches, they will come in singles or sets (you will not need anything larger than 3mm in almost any coil build I can think of.
Stainless steel mesh comes in a variety of mesh counts; 100% PG juice works on the 500 mesh and mesh in the 300 range is VG friendly. If you have yet to develop a preference for your juice VG

G ratio, get an assortment of mesh sizes. A sheet of mesh will make a lot of wicks.
As far as Kanthal goes, I just use 32ga; if I want lower resistance I wrap multiple strands together (YouTube coil building videos and you'll run across how to do this). Otherwise there are a few different wire sizes of Kanthal, each have their own qualities. I feel comfortable with just the 32ga some folks aren't.
Things like tweezers and wire snips, again a tool shop will likely have these or you can use fingernail clippers and a pair of not so perfect tweezers from the junk drawer.
Small screwdrivers, buy the proper size screwdriver is all I can say here. There is no better way to jack up a screw than by using the wrong size driver (yep its a silly pet peeve of mine). If your screws are stainless steel or made of a non magnetic material don't bother spending extra on a magnetic screwdriver, but if it doesn't cost any more buy the magnetic version.
You'll want an ohms and volt meter, if your device doesn't already function as such.
Maurice