The first triton test is so you know where your numbers are "before" you start with additives. This lets you know where heavy metals might be in your system, if you are lacking something or if perhaps you have some discrepency between your test kit readings and what the ICP test states. You also want to make sure that your Alkalinity is as close to 8.0dKH as it can be PRIOR to starting dosing of Triton Elementz.
For Triton Elementz - it is recommended that starting dose is 10ml of each element (1, 2, 3a & 3b) per 100l of tank water per day. You should perform Alkalinity testing EVERY DAY for the first couple of weeks. This is so you can alter your dosing (the amount of elementz added per day) to maintain an Alkalinity of 8.0dKH. As you add more corals, or they grow larger, then you will generally see an increase in the uptake of Alkalinity (meaning that the value will start to drop below 8.0dKH even at the same dosing amount as you did the previous day) and you will have to increase your dosage.
I wait a month before sending out my second test. This gives my system time to stabilize using the Elementz, but close enough that I recall anything that might have happened between my 1st test and my 2nd test (new equipment, dropping something metallic in the tank, etc etc etc). Once you have your second set of results, you can adjust dosing as needed. Keep things stable, and in another month send out another test. Once you have three sets of results, you can start to identify trends. Is my Aluminum rising? Do I need to dose Iodine? etc etc etc...
From there its about testing frequently (I kept testing Alk daily - but with the Hannah checker it takes about 2 minutes) - maybe once a week or so - and keeping up with the demand that your corals and livestock will want for dosing.
As for cost - Triton Elementz runs about $70 for 10l worth of dosing solution. For my 300 gallon system, that would last about 3 months. Zeovit has some inherent costs as well (Reactor, zeolite media, bacteria, bacteria food, and other supplements), but I think in the end its likely a wash. For information on the zeovit system, I would suggest watching the
bulk reef supply 52 weeks of reefing series, as they are using the zeovit system on that tank, and go into detail about their system and what and why they dose. Its pretty informative about that system.
And, as always, feel free to ask on the forums here - there's a wealthy of experience in both of these methods, and Im sure more people will chime in with their experiences.
Good luck!