Great question - its something I recently have been addressing myself. After speaking with the "lady at the fish store" (as I refer to her in colloquialism), she recommended the following with the following justifications:
API for Ammonia - because it is a much quicker method of testing, albeit less precise
Hanna Checkers for phosphate, and alkalinity - because these are tested routinely at the store and the devices are the epitome of "fire and forget" easiness (despite having to reload the reagents)
Salifert for calcium, copper - because although it is more of a hassle, getting the copper cycle rebooted is much more so, so you will want to be as precise as possible with your copper readings
Red Sea for everything else - this I imagine was more of a sales ploy than actual advice....
Despite the fount of knowledge, I use a different approach. In part, it is because I went a (slightly more) cost effective way (being a poor law school grad ha!) But also, it is the product of habit and availability. While part of me wants to be as precise as possible with my chemistry, I am just not quite there yet in my finances department.
I have Salifert for Mg, I2, and Ca. I also picked up an API saltwater test kit and an API master reef test kit to cover the basics - i.e. ammonia, nitrite, nitrate, phosphate, pH, etc. (about $45.00 for both API kits.)
The takeaway, I guess is "it depends" (he said in classic lawyerly fashion.) I was advised that Salifert is great but Red Sea is better. I cannot say that this information is the best available but it has helped me
(P.S. I would do dirty, dirty unforgivable things for a full set of Hanna checkers - man, those things are so slick but I feel you on the price-point issue.)