A lot of this depends on your live stock list and bioload.
If your tank is a FOWLR with shroom and soft corals then Ca isn't as important as is a heavily populated SPS tank.
The question shouldn't be what do you test for, but why do you test?
Ammonia - (0.0) Once the nitrogen cycle has established you should never see a spike in ammonia. Just make sure you pull out any big stuff that dies.
Nitrite - (0.0) same as ammonia
Nitrate - (<0.2 ppm) The end of the nitrogen cycle, unless you're running a plenum. Test this and when it rises increase water changes. If you can't keep it low, your tank is over it's max bioload. Either take some fish out or add a bigger refugium.
Calcium - (400 -500 ppm) Used by corals and coraline algae to build their skeletons.Not enough and they can't grow (or at least not well). To much and the pH becomes a problem.
Alkalinity (2.5 - 4 meq/L or 7 - 11 dKH) Acts with the calcium to set the pH. Also (I believe) aids in the uptake of the calcium by the animals / algae.
Salinity (35ppt or 1.026sg) - Suprised no one mentioned this, cause I know you all test for it.
Temperature (76 - 83` f) - This also depends on the livestock you're keeping, but for the majority of us it's tropical fish / inverts. Again, this is a test you all do frequently, yet no one mentions.
PH (7.8 - 8.5 [8.1 - 8.3 prefered]) - How alkaline or acidic the water is. (pure water is 7.0, above that is acid, below is alkaline). This will show you if the relationship between your calcium and alkalinity is off. Typically not needed if your other parameters are set corectly.
Magnesium (1250 - 1350 ppm) Alows the calicium and alkalinity to interact. Not enough and the alkalinity will not allow the calcium to precipitate, thus not allowing the animals to uptake it.
Phosphate (<0.03 ppm) - Algae (the bad kind) food. If you use RO water and good quality salt you shouldn't have any. Also, macro algae in the refugium uses it faster than other types keeping it in check. Unless you have algae problems don't bother. If you do, check you raw water (that you use for top off's and water changes) even if it's RO as this will show a bad filter. Check your salt water mix prior to use and this will show if you should maybe think of getting a different brand. As a side note, phosphates are used in soap, so be sure anything put in the tank (like your hands) are washed throughly and then RINSED WELL.
other items, silicia, iodine, strontium boron, iron are used sparcely by the animals and are generally kept in check by water changes. If your tank goes outside what is normally done in this hobby, then look into these. For example a very high stock rate of inverts may need iodine suppliments because they need it to molt. very high coral stock may need extra strontium (they use it in conjunction to builb their skeletons).
So, for me, the things I check are...
Salinity
Temperature
Magnesium
Calcium
Alkalinity
Nitrate (occasionally)
If the tank is having problems or I don't like what I see going on in therre I'll test other things. What I test depends on what I see.
