Wow, thanks for all the info! As usual, so much that I did not know. So I really shook up my 2nd test bottle for the nitrate test this time (thanks, Shawnie!), and . . . I discovered that I do in fact have nitrates! Between 5 and 10 ppm. I'm embarrassed to say how excited I was about that! :-)
Thanks for the info about the test kits, Lucy. I am using a salicylate (sp?) test -- the
API master test kit. So I understand that I may be picking up a false positive from using Prime. I went ahead and did another water change anyway just to be on the safe side but didn't test for ammonia again. I realized after my kids went to bed (the fish are in their room) that I should have tested right away for ammonia, in order to get an accurate test. Oh well, too late for tonight.
jdhef, I did the cycling method where you add ammonia in the very beginning. I got it from aquariumguys.com and it was helpful but I wish I'd found the one on this site first. The one I used didn't mention testing nitrates. However, it appears that I do have them now!
eignh, my tap water tests completely negative for ammonia, thanks.

again, everyone! I will keep testing and hopefully my tank will get balanced out so it doesn't need so many water changes.