You are correct in your assessment of API. IF the tank has been set up with water for a couple of months the cycle is done, right Brandon?!
You don't need kits to keep water, but if you want fish and coral down the road what you want to keep will tell you what you need to test for.
Salinity, temperature, are basic for all life. The fish don't need you to test if you keep up with water changes they generally can survive what we throw at them.
Coral are different. They like steady dKh, Calcium, and Magnesium (often called the big three)
Salifert dkh is accurate and not pricey, I use Red Sea for Magnesium, and Calcium I can't recall the crazy cheap test I've used that is good enough for me because I have a calc reactor that keeps dKh and Calcium stable so if dKh is solid I know my calcium is rocking. Sorry I can't recall the calcium test. Someone gave me the Hanna Calcium, and I have Hanna checker for dKh and ULR phosphorus and if you get into hobby craziness you might spend extra, for the Hanna Checker ALK ANd Phosphorus later, based upon the answers to these same questions that a few folk post here each day.
Glad you asked, we all gotta help each other along with this hobby.
@brandon429 -Happy New Year to the Sand Man!