Yes use Prime. Prime takes out Ammonia @ .25 ppm of ammonia and won't convert anymore to ammonium. Your water contains a lot of chlorimine and thus you have high ammonia readings for your tap water.
What you are probably getting is ammonium in there (half and half of ammonia and ammonium NH3 and NH4) the testing kits won't differ NH3 and NH4. So if you are getting 1ppm of ammonia then probably in reality it is: NH3 is .25ppm and NH4 is .25 ppm and you are safe to do your water changes with your tap water because once your tank is cycled the .25ppm of ammonia will be converted to nitrates and will be absorbed by your floating plants. :) No problem man just keep your cycle going and your fish will be fine.
Do not carbon dose your tank just add floating plants the less chemicals used in your tank the better. I only use Prime in my tanks nothing else.
For Nitrates I think you can add plants. Even better floating plants! Duckweed, salvinia, frogbit, and other floaters is what you want to suck those nitrates right out. They will take up the extra Nitrates just don't let it cover your entire tank or it will be dark. Just take some out especially for the floaters that grow in large amounts in a few days like duckweed.
Ammonium is less toxic than ammonia, but still toxic. The beneficial bacteria, which are two kinds, do the following:
One eats the ammonia and the other breaks the ammonia down into nitrites. The nitrites are further broken down into nitrates which get sucked in by plants. Which is why it is called a cycle. Not sure if this makes sense, but you get the idea.
Good luck with your aquarium :)
:) Yea we all have this question... then we learn about NH3 and NH4 :) and that test kits don't read the difference between them.