I think you should try to cycle the tank you have now! Is it a five or ten gallon? If you're able to do this, you'll be able to neutralize the ammonia in your tap water (it will still show up on your tests, though*) long enough for your cycle to convert it to nitrite which then, in turn, gets converted to nitrate. That would handle your ammonia problem nicely, methinks. There's also Seachem Prime water conditioner that neutralizes ammonia, but I'm not sure how long, or how people use that in uncycled tanks, because eventually it becomes toxic again, right? I have no ammonia in my water, so I really don't know how people get around it :<
*You say it still shows up on your tests after treatment, but that's because even after the ammonia is reduced to ammonium it is still detectable. The difference is that ammonium is not harmful to your fish (I think), but the problem is that it doesn't remain in this state forever and will eventually go back to ammonia (I think).