Despite my posts, I am a relative newcomer here and have only owned one test kit, but I got it because it is the one most often recommended here, its the API Freshwater Master Test Kit. There is a Hagen Mini-Master that is also good but lacks a Nitrate test I believe. I believe there is a Red Sea one for freshwater but I don't know which of their kits it is.
You definately need Ammonia and Nitrite tests. Next most important are pH and for folks with very high pH there is a High pH test sometimes needed. Then Nitrate is nice to have. These are the ones in the API collection or can be bought individually if you run out of a particular one. People who have trouble with low pH sometimes end up going out and getting a KH test because "Temporary Hardness (KH)" is highly related to this.
These are all liquid-reagent based tests and are the appropriate level of accuracy and cost for most of us. Test strips are considered here at TFF not to be worth the paper they are printed on and tablet-based ones, for the tests other than pH, have also been not thought too highly of. On the other end of the scale, there are kits or instruments that have higher accurace than what we use but are overkill.
The topic of exactly what ammonia to find and get is much discussed and depends a lot on where you live - thus one of the needs for people to put their location in their profile. The search button up top right can help you find many of these discussions.
~~waterdrop~~