It's a delicate operation. It generally requires tools the average DIYer may not have access to.
The AWG32~38 cable typically used for tonearm wiring is difficult to work with. Start with a bundle way longer than you need; you will be shortening it as you fail.
A solder pot is useful for tinning the wires. Tinned wire is significantly easier to deal with than bare wire. You can also use a solder pot to remove any dielectric (eg teflon) or with magnet wire or Litz cable, the chemical coating that serves in place of a plastic dielectric.
Instead of a solder pot I use a ceramic crucible and a butane torch. I use Caustic Soda (Sodium Hydroxide) which is a dangerous base cleaning agent to remove lacquer (etc) chemical coatings when necessary. WEAR EYE PROTECTION.
You need a temperature-controlled solder iron (HAKKO 936 or equivalent) with the corresponding fine tip and the finest solder you can get. You also need electronic-grade flux, you can't rely on what is incorporated in the solder alone to do the job.
It is a mistake to go lead-free here; the temperatures required aren't conducive to a successful job.
Also, I believe STAX uses a non-standard configuration at the tonearm end for connection to the phono connector (often referred to as a DIN connector, but it isn't technically DIN). So be sure you are attaching the right wires to the right point and that the gender of the connector is correct.
George Cardas has some tips on his website about dealing with his tonearm rewire cables.
