Hi. I'll try to offer some advice. It's good that you know about ammonia. Ammonia comes from fish respiration, waste, and rotten/ uneaten food and plant matter. Good bacteria grow (primarily in your filter material) and convert the ammonia to nitrIte which is also toxic to fish. The nitrIte is then converted to nitrAte which isn't of much concern until it gets to very high levels. What you need to do is get test kits for ammonia, nitrIte and nitrAte. I would test the water daily for ammonia and nitrIte. If you see the levels getting to 0.25 or higher then you should change some of the water (10-20%). When you change the water, vacuum the gravel to suck up any uneaten food and waste. I would also test for nitrates on a weekly basis. Ideally you want to keep nitrates low as well, (below 20ppm; lower is better) keeping in mind that you won't see nitrates until the tank has cycled. If you monitor the water daily, you will see the ammonia rise, then fall, then the nitrItes will rise and fall (both to zero), and then the nitrates will rise and you control that by doing water changes. This whole process will take a matter of weeks, not days to accomplish so if you wish to keep the fish then you must be committed to the maintenance. Once the tank has cycled, you can back off on the testing and water changes to weekly or bi-weekly. Also keep in mind that neon tetras are not known for being the best fish to cycle a tank with. Water changes are your friend. Don't over feed. Don't forget dechlorinator and try to keep the fresh water added at the same temperature and pH as the old water. With some luck the fish will be ok.
As for other fish, you are limited by the size of your tank, but there are many options. Neon tetras, as many fish like to be kept in groups of 5+ so i would add a couple more to your group when the time comes. Other than that, you can add almost any other tetra that doesn't get too big, harlequin rasboras, corydoras catfish, guppys, mollys, platies etc. etc. Research the fish first to make sure they won't get too big. You are limited to about 20" of fish (total length minus tail) of the fish's adult size.
Good luck with your tank... I'm sure you'll be ok in the long run. If you have more questions feel free to ask. I would be happy to help as others would also.