The first thing you do is cycle your tank. For that you need ammonia, nitrite and nitrate. I would not advise the API tests because the ammonia test gives a low ammonia reading when the actual reading is 0. And you do not want to put livestock into a tank that still has ammonia (Highly toxic),
I used Red Sea starting out. I now use Red Sea Pro (nice thing is you can get replacement chemicals so you do not have to buy the hardware.
Tests you should have
Ammonia - I used this in cycling and have not used it since.
Nitrite - I used this in cycling and have not used it since.
Next are the nutrients: nitrate and phosphate. Too much and you get an algal explosion and too little you get cyanobacteria or your corals starve.
Nitrate - Salifert, Red Sea Pro, and I hear the new Hanna High range is good.
Phosphate - Hanna is the only one that makes a sensitive enough test to be useful. Use either Hanna ULR phosphate or ULR phosphorous. The ULR phosphorous is a bit more sensitive.
Coral building chemicals (also shells for marine mollusks). I have an small polyp coral tank (reef building corals). I have a doser for ALK. I hand dose Ca. And Mg seems to do fine with just water changes. I have a fair bit of growth but if my growth was extreme, I might end up dosing Mg.
Calcium - Red Sea Pro. I have not tried Salifert but I would think it is good.
ALK - Red Sea Pro, Salifert, and Hanna. Hanna has a big reputation because it gives a number. On Red Sea Pro and Salifert, you have to watch for the color change at the titration end point and that feels a bit like a judgement call. I regularly use all three. Oddly enough, Salifert (the cheapest) might just be the one that varies the least (most precise),
Magnesium - I use Salifert. Red Sea Pro drives me crazy On this test.
Trace Elements - there are tests for trace elements. I have tried a few but they did not inspire confidence. You can send off a sample of seawater to get tested for trace elements but that has issues also. The best way to keep trace elements in line is to do regular water changes (in my experience and I do not consider myself to be the last word on this.