ammonia, nitrite nitrate and ph are all pretty important. phosphate is a nice test to make sure you dont add an algae problem (like I did)-although some research has shown the bacteria that converts nitrite to nitrate multiplies faster when phosphate is present. someone tried to help my tank, added tap water and my levels went to like 20ppm-so ill have algae for a bit of a while as i continue to add rodi
when you add corals calcium, magnesium(I dont think corals do poorly whatever the level is, but they do better with it) and carbonate hardness.
I use a chlorine and a copper test once in a while on my water just to make sure nothing bad is going in. I also test the water i get from the lfs
to make sure i dont need to get them out of the bag water asap.