My approach to testing is changing a bit following an algae outbreak despite the fact Nitrates and Phosphates were showing zero in tests...
therefore:
When starting up a tank (and for the first few months) you need to monitor Ammonia, Nitrites and Nitrates to know if the early cycles have completed.
Once it is up-and-running, the tank will tell you if you have an issue with Nitrates &/or Phosphates (where tests will still show zero). This is because the tests can only detect the presence of these in the water - algae is capable of taking these out of the water => they are not detected. So what use is testing??? well if you see algae, you can at least tell if you have excess Ammonia, Nitrites, Nitrates or Phosphates, but the algae will show first. => I will test these periodically (1/mnth) or if I see algae start to increase.
On-going I feel it is important to maintain:
Alkalinity (the waters ability to hold stable pH)
pH
Calcium (particularly for corals)
Tests for elements such as Iodine can also be misleading, depending upon the type of iodine you use, for example Kent produce Tech-I which has a built-in slow release mechanism for iodine. Tests can not detect this unreleased iodine, so may lead you to add more when you don't need to etc...
Agree with temp and salinity (seen many posts by people who have topped up evaporation with salt water only to be surprised when high salinity has killed something in their tanks).
that's my :twocents: