Hi! I see Byron is helping you, and you lucked out with that! He is very knowledgable! But I thought I'd help ya out for now, since he might not be back on until tomorrow.
I hate to be the bearer of bad news... but the Peacock Fern is not a true aquatic species and will slowly die if left submerged. So since it is doomed, its not going to grow well enough to help grab up the ammonia. If you're able to return the fern, I think that is the way to go.
While you're there returning it, or when you get to a LFS (local fish store), you'll want to pick up an API Master Kit (like Byron suggested) with which to test your water parameters. Until you get that, I would be doing daily 50% water changes with a good water conditioner!
Its also good to feed sparingly for right now. Are you already doing daily water changes? Do you have a filter on the tank? Do you have a heater?
If you think that it might take you a while to find some good plants, there's something (in addition to 50% water changes) you can do in the meantime to help out your little guy- Do you know anyone that has a HEALTHY tank (no diseases, excellent maintenance, NOT from a fish store) from which you could borrow a handful of substrate or a used filter cartridge? If so, adding this to your aquarium will introduce some beneficial bacteria that will jumpstart your cycle and help make the water healthier for your fish while you're getting it planted. To do this, you literally take the "seed" from one tank, keep it wet in water from the tank, and put it right into yours. The seed has to stay wet with tank water, can't touch tap water, and has to be transferred as quickly as you can. If you can't get plants right away, this will help!
As far as a 15g column... this shape will be limiting for you. The more surface area a tank has in relation to volume, the more fish you can put in it because the water has more room for oxygen exchange. Fish also have space needs due to activity level of a species, or territorial issues. For example, the stocking level for a 20g tank would be the exact same as the stocking for a 25g tank because the surfaces area is the same, they're just different heights. As a general rule, its almost always better to chose the tank with the largest surface area possible.
You're on the right track coming here! There are many very helpful people here. Pretty much all of us will tell you- don't listen to anyone from a pet store. We're here to help, and we're not trying to turn a profit. We just want to help you and your fishies!