Heat shrink tubing isn’t totally necessary. It’s just for wire management, which, when you’re starting out, can look quite terrible.
I prefer the copper braid for desoldering mistakes. Pumps or suckers tend to shake when you use them, and it’s a little harder to control. Also, be sure that if you use copper braided wire, hold it with pliers. Remember, it’s conductive, so holding it directly on to the hot solder will conduct it up to your fingers.
Wattage is something to consider. How much heat do you want is what matters. Higher is not necesarrily better, especially if you haven’t had any experience, because you can toast things to a crisp. But higher heat allows you to work more quickly.
I’d recommend picking up some perfboard while you’re at it, and then just solder wires from point-to-point. Try going through the holes first, but then try soldering to the surface, because you will have to solder to the surface with your padhack.
Leaded solder. It’s cheaper, and it works at lower heats better than its lead-free counter parts. Get the kind that is listed at “Light Duty.” That’s as thin as you want it. Any thicker is just a pain. 60/40 rosin core is the norm.
Stranded wire. 22 gauge is probably the smallest you’ll find. Solid core is terribly hard to work with, and it will break right off easily. Also, a little tip about wire, tin the wire, that is, melt a little solder over the wire strands to bond them together, then use this solder that’s on the wire. Melt again when you get to the contact you want to solder to, then once it flows against the contact, remove the iron, let it cool, then it will be soldered in place.
Also, you’ll need these: RadioShack? 8.5" 4-Way Crimping Tool - RadioShack.com They are for cutting and stripping wires, and they also crimp quick disconnects.
You don’t need quick disconnects. If all you did was take your PCB out of the SE, then you still have all the quick disconnects that came with the stick there.
Also, you may need a very tiny screwdriver to open up your fightpad, and a larger one to open up your stick, but I assume you already have those.