In summary, the most important test kits to have when owning a saltwater aquarium are:
Ammonia – This is needed in the very beginning, but will probably never use again unless you decide on running quarantine tank.
Nitrite – Also needed in the very beginning, but will probably never use again.
Nitrate – Used to check that the nitrogen cycle is established, and then weekly to monitor levels.
Phosphate – Used weekly to monitor levels.
If you decide to keep corals:
KH – Used to check alkalinity levels weekly.
Ca (Calcium) – Used to check calcium levels weekly.
Mg (Magnesium) – Used to check magnesium levels weekly.
There are so many different choices of test kits out there and each brand has its own dedicated uses. My preferences are Salifert and Red Sea, both of which I have found to be very reliable.