Tools and supplies you'll need:

A soldering iron

Mine happens to be an adjustable one, but the kit doesn't require it. A regular fixed-wattage iron will do fine.

You do need to throw some heat around though.. header pins have a lot of mass compared to the leads of a resistor, so it's harder to get them up to 350C (the temperature at which solder melts). If your iron is too small, the whole pin will heat up slowly, and you'll end up melting the plastic around it long before it's hot enough to melt solder.

Solder

The ROHS (Reduction of Hazardous Substances) standards require a shift to lead-free solder. It keeps the heavy metals out of the landfills, but as of 2010 lead-free solder is a bit harder to find than plain old 60/40. It also costs more, and is a little harder to work with.

Reflow solder paste is almost exclusively lead-free, and industrial fabrication uses a lot of reflow because it's faster and easier for SMT packages than per-component soldering. For the hobbyist with a soldering iron, odds are that you'll probably use up your existing roll of solder before moving to lead-free.

Lead-free is the way of the future though, so you may as well get a roll and start learning to work with it eventually.

A file

Breaking the circuit board into sub-boards leaves ugly rough edges. You can leave them as-is, but they'll look better (and snag less) if you clean them up.

Tape (optional, but highly recommended)

When you install the 2x4 patch headers, there's this inconvenient "flipping the board upside down" step between the "putting the pins in the holes" and "holding everything in place with solder" steps. If you aren't careful, that can introduce "throwing parts all over the place" and "disasembling your entire workbench to find the part that random chance threw into a spot you couldn't reach deliberately with 10 minutes of work and a hammer" steps.

Speaking from experience, those add nothing to the joy of hardware hacking.

A bit of temporary support from tape can save you several minutes of flexing your vocabulary.

A soldering pencil (optional)

This one is purely convenience.

From a functional standpoint, the pencil is a highly-evolved interface between the human hand and a long, thin, consumable filament. A mechanical pencil inherits most of that design wisdom, then adds a layer of engineering to let the filament move independently of the carrier on demand but stay fixed when necessary. Those designs are much younger than the pencil per se, but have still gotten pretty darn good.

Solder is a long, thin, consumable filament.

It seems a shame to ignore that much accumulated knowledge when you can get it for a buck at WalMart.

A trashed breadboard (optional)

I come from a mechanical background (carpentry, cabinetry, luthiery, machining metal and plastic, toolmaking), and about 80% of the training in all those fields involves learning to use jigs and fixtures.

All work requires some amount of force, and very few fabrication processes get easier when you're trying to hit a moving target. I never hold a part by hand if I can find a device that will do the job for me. My hands have more complicated things to do, and a fixture almost always holds things better than I can.

A breadboard makes a great fixture for things with pins on .1" centers (like pin header), but doesn't take heat very well. There's always a chance it will melt a bit if you use it to hold parts while you're soldering, so don't use your best one.