Nitrite is good to know when a tank is done cycling, or if you are having die off's, its a good thing to test in addition to ammonia. Nitrate as well should be test time to time to know if a WC is in order. I've never bothered with copper/calcium/phosphate before.
The API master kit comes with ph/ammonia/nitrite/nitrate and then get the gh/kh and should be good really. Watching the ammonia/nitrite/nitrate is really the only way to know if a tank is fully cycled or not as you should be able to follow a graph like this.
to really know whats going on with the tank and params. Sometimes with established media and a good source of ammonia, a tank can cycle in a week or two. Other times it can months and without readying all 3 of those, you will really never know for sure whats going on with it as you can see from the graph, ammonia may be almost zero and you think its done but you're in the nitrite phase instead and no where near doing being cycled. Like I said, its good if you start to have a few die offs to test all 3 and see whats up.