The most important things you need to test for are the nitrogenous wastes: ammonia, nitrites, and nitrates. You want ammonia and nitrite levels to be zero, and nitrate levels to be below about 40 ppm. Ammonia is produced by fish; certain bacteria feed on it and produce nitrite as a byproduct. Other bacteria feed on nitrite and produce nitrate as a byproduct. These bacteria may take a little while to become established in your pond, so it's important to test frequently when your pond is new. If levels of any of these are too high, run dechlorinated tap water or well water into the pond to dilute the wastes.
Nitrogenous wastes are seldom a problem in ponds with healthy plant populations, as plants absorb all kinds of nitrogenous wastes. Ideally you want your plants to be established before you add fish.
You want pH to be between 5 and 8 and relatively stable.
All the other things- hardness, phosphorus, and so on- make little difference to goldfish so long as the levels are stable and not too extreme. Too much phosphorus can encourage algae growth, but your plants should help keep that in check.
