As I said in the introduction, guitar setup work does not require any fancy tools. I'll tell you what
you need as we go along. You may have to order some simple stuff like a set of nut slot files from a
lutherie supply store such as Stewart-MacDonald. That will
take some time. The important part is to be patient and not go at it with inappropriate tools.
We've all done things like trying to tighten or loosen a screw with a kitchen knife or a coin. Remember
how you regretted that? Don't do stuff like that again, especially not when your guitar is involved.
There is one tool that perhaps bears some discussion, and that is the set of automotive feeler gauges that you'll need. Any standard set with a range of something like .002'' to .035'' will do. If your set is metric, the thickness of the blades will be given in millimeters. To convert from my numbers (which are in inches) to millimeters, you'll have to multiply by 25.4.
You will be using the feeler gauges to measure the gap between strings and the frets below them
in various places and under various circumstances. Oftentimes, you will be looking for a gap
that measures something like .090''. To get that, you will have to find several feeler blades whose
thicknesses add up to .090'' and stack them on top of each other. For example, a .025'', a .030'', and a
.035'' together will make a .090'' blade. Therefore, it is essential to find a set of gauges that can
be unscrewed like the one in the photograph above, making it possible to use the blades individually.
If you can't find one locally, you can order it online, e.g. from
Auto Body Toolmart.
When measuring a gap between a fret and a string, you want to use what mechanics call the go/no-go method. Find a blade (or stack of blades, as the case may be) that slides easily and comfortably into the gap. Then increase the thickness of the blade in small increments until you find the first one that does not fit into the gap anymore. The true width of the gap lies between the last blade that fit and the first one that does not fit anymore.
Needless to say, when measuring a gap underneath a guitar string, the no-go in go/no-go is not really a no-go. Thicker blades will go into the gap quite easily, but in doing so, they will push up the string. So really, your first no-go blade is the first one that does not go into the gap without pushing up the string. It takes quite a bit of concentration to watch the string for even the slightest movement as you slide the blade(s) underneath it. Also, it takes a steady hand to get the blade(s) underneath the string at just the perfect angle without wiggling, turning, or angling. You'll get the hang of it.
When used correctly, the automotive feeler gauges will give you sufficient precision for measuring string-to-fret distances. However, people have also reported excellent results with more advanced tool sets such as machinist plug gauges with a micrometer.
Finally, Stewart McDonald sells a string action gauge, which is a special ruler for guitar setup that allows you to read things like string action on a scale. Being engineering-minded, I will probably stay with the go/no-go method. But the ruler is clearly more convenient to use, and the accuracy may well be good enough.