Salinity, pH, water temperature, ammonia, nitrite and nitrate for the basics. From there you would test calcium, magnesium and alkalinity. Lesser areas to test, however, can still be important are things like CO2, O2, silica, phosphates, iodine and light intensity.
This is a pretty good list. Depending on what you have in your tank will have some bearing as to how many of these you will need to measure and monitor and how often.
Usually, the rule of thumb is if you are dosing it, you check it. Obviously, if you have very few hermatypic corals then testing for calcium, magnesium and alkalinity are not going to be as important. Nor is it pertinent to check your CO2 level if you don’t have a reactor.
This is just a guideline...