Whe adding fish to a new tank, think about this.
The 1st fish goes in. You have cycled the tank meaning that there is a population of bacteria equal to the bioload that you had in the tank, ie equal to a rotting shrimp, or whatever you used to cycle the tank. The bacteria population is in equilibrium with the bioload, which is slight. Next, you add a fish. The bioload doubles or triples. Suddenly, the bacteria cannot keep up with the bioload. Ammonia rises. Nitrites rise. The bacteria population grows, and slowly consume the ammonia and nitrites. Then the bacteria are again in equilibrium with the bioload. The next fish added now increases the bioload, maybe doubling it or less this time. The bacteria again are behind, but can catch up faster because there are more to start with and because you have upset the bioload by a relatively lower amount. The next fish will only increase bioload by 33%, and the system responds even faster. Eventually, you can add a couple of fish at a time and the tank will hardly notice. This is an established tank.
Your tank is new, and you need to slowly add fish, one at a time to get the tank established. I usually add a fish and wait around 6 weeks til I add another fish. And I don't add if the other fish are sick or not eating or not healthy. Once the tank is established, you can add more than one fish at a time, or add them more quickly.
When the tank gets upset, high ammonia, nitrite or whatever, chances are that your fish will get sick or die. The advice above on quarantining and ich is great advice.
Tangs are very prone to ich.
Buying healthy fish is also very important. I always make sure the fish is eating in the LFS tank. Also, I look closely at the transparent fins and look for spots or blotches or fin damage. These are all signs of disease - ich, velvet, fin rot. I also watch them breathing. Fast breathing is a bad sign. Scratching is a bad sign.
I use copper to treat ich. It works. Most other ich cures out there do not work. There are other non medicine ways to kill it, but I find that copper works just fine for me. But it will kill all inverts.
I hope this helps.