For a reef tank it's good to test for Ammonia, Nitrite, Nitrate, pH, Phosphate, Calcium, Alkalinity, and Magnesium.
In softy only tanks alkalinity will be harder to manage. In hard coral tanks calcium will be hardest to manage but alkalinity will be used as well.
Magnesium doesn't change much. However, it's good to keep an eye on.
Sadly, the API test kits are next to useless for a reef tank when it comes to phosphate, calcium, alkalinity, and magnesium (I don't even think API has a mag test kit) I would suggest either Hannah checkers or the Red sea test kits.
Mebbid's Fishtopia: Various builds
"The simplest explanation for some phenomenon is more likely to be accurate than more complicated explanations." -Occam's razor