Like others have said there is no 100% way you can know how fish are going to behave when a new fish is added to the tank. Even fish that are listed as being friendly and compatible according to the charts may fight (its all just depends on each individual fish's personality).
Having said that, I also have a Yellow Tang, a Yellow Tail Damsel, and a Diamond Back Goby and they all get along just fine. I wouldn't worry too much about the Damsels and Tang fighting with the Diamond Goby, since the Goby will spend most of it's time on the sand while the Tang and Damsels swim higher up in the water column.
If you want to maximize your chances of a peaceful acclimation, when your ready to add the new fish to the tank, feed all the fish thats already in the tank and turn off the lights, that way all your fish will be full of food and sleepy when you put the new fish in.
Even still the fish in your tank might try to assert their dominance over the new guy, but that's normal and should come to a stop in a week or so.
