That part about backing out the pins is an important point. Since zither tuning pins work by screwing into the wood, if you just cut the existing string and start from there in tightening the new string, you end up with a pin that goes from 1/2 way in to one that's 3/4 or 7/8 of the way in. Back it out 3 whole turns before putting the new string on. Then you're back where you were when you finish, because you should have 3 turns or so on the pin to prevent breaking the string from metal fatigue.
If you're going to do all the strings, you can back all the pins out to about the same position (use a straight edge to find any outliers). It not only looks nicer when finished, but makes tuning easier -- a pin that's too low is harder to get the tuning wrench onto than one that's the same height as its neighbors.
And, to save yourself trouble, get a good tuning wrench -- one with a star head, not a square one. Look for zither tuning wrenches (or even harpsichord tuners -- they're the same beasts). Those wrenches that come with autoharps are usually pretty bad, and can chew up the tops of your tuning pins.