Damsels are highly terrirorial and unless you want to keep just one, there's no way you keep 2 or more in a 10 gal tank. Even if they're the same species. The limited space will make them aggressive. In a larger tank, they can go to their own territories/ homes if they're chased by a fish.
Bullying occurs because they want to protect their territory, but in a nano tank, damsel often takes it to the next level and they will not quit chasing the fish until it's dead. But if you plan to keep just one, go ahead they're colorful, hardy, and are not choosy about foods.

I recommend two fish and no more. Ten gallon is really not much space for marine fish... even for gobies.
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Even if your tap water doesn't contain chlorine and chloramine, it still contains metal and phosphate that are still dangerous. Best to go with is the R/O water you can purchase at LFS.
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I think for a beginner, it's better to go with live rock and not fish only tank. It's better to have a biological filter. I think it takes someone experienced to completely rely on the water quality on mechanical filters because they don't really filter
much of the waste in marine setups.
Biological filter should be the main system and helped by the mechanical filters. You should also use protein skimmer. For a nano tank, a small one that uses air stone should be a good fit... if you use the common bigger skimmer, it's too expensive and will cause too much water current.
But never go with undergravel filter because it will never work in marine setups. You can research more why that is... I find it appalling some books still suggest undergravel filter for marine, because it's totally so wrong. Honestly, I forgot the reason why this can't be used in marine setups, but all I know is you just can't lol
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Water quality... in addition to what koko said about weekly water change... you should always monitor how much water is lost to
evaporation.
In fresh water you could get away with evaporation, but in marine setups it could cause death. When water evaporates, the salt remains, this means the tank gets saltier. You could put a marking with a marker or tape to know where the water level is when you first set it up so you know how much water is lost and how much you should add. And since the salt remains in the tank, for water top off, you only need to put in just water (without salt).
Also before every water change, it's important that you do it when
you have finished topping off the water lost to evaporation. If you do water change before you top off the water that is lost to evaporation, what happens is you're getting rid of the saltier water in the tank with water with normal salinity that you do. You may get away with this once or twice but over time, over 4-6 months or more, this will considerably reduce the salinity of the water in the tank, especially considering your tank is a nano.
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Clean up crew will help you a lot. Get some turbo snails, hermit crabs.
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Tank set up should be with live sand to go with the live rock... over time the beneficial bacterias (hitchhikers from live rocks) will reproduce and they will host in the sand. This is why the thicker/ higher the sand the better... but make sure it's not too thick you can't enjoy your tank. You will see thousands of critters (pods) on your tank glass... they're the good guys. And once you add a fish like damsel or clown, they like to eat them so the pods will burrow and live in the sand. They're helpful in processing the highly toxic "nitrite" to a still toxic but much more tolerable "nitrate".
I believe the perfect set up should read all ammonia, nitrite and nitrate at 0.
I think so far that should be a pretty solid basic... there's plenty more to learn, including lighting, sump, calcium, kH, pH and other advanced level care, which I'm not there yet lol
I hope you enjoy should you decide to give it a go

Remember patience is gold with marine tanks...