If we don’t include the obviously most important parameters of salinity and temperature, then for a fish only system I’d say nitrate and phosphate are most important. However if you’re going to be keeping corals you might as well get into the habit of testing alkalinity, calcium, and magnesium as well.
With a fish only system none of the parameters are nearly as important as they are for corals, and even the two I mentioned have more to do with avoiding algae issues than the health of the fish.
For nitrate and phosphate I strongly recommend Hanna checkers (ULR for phosphate and HR for nitrate). I would start testing them about twice a week then after a while I’d probably go to once a week.