Dana, first let me say that (in my opinion and I'm definately biased by my own problems, lol!) I think plants are a hard thing to learn, quite a bit harder than fishless cycling and that one's not easy. So you have to look at it as an ongoing learning thing, perhaps over the next year or two, its that kind of timescale I think on average.
Sounds like you've got about a half watt per U.S. gallon (.5w/g) and that's pretty low, possibly off the low end of the "low-light technique" range, but I'm not experienced enough to quite know... it might be ok for low-light technique. With low-light method you need to choose from those species of plants that can handle low light, like java ferns, anubia nanas perhaps, perhaps some amazon swords, I don't know what else (George Farmer has a post somewhere on "easy plants.") Now low-light technique itself is usually a -good- thing for beginners because light is like the "gas pedal" for tanks with live plants, increase it and lots of things start happening fast and the chance of accidents gets greater.
Another thing I see in your situation is that if your mom is right and the water is soft and your pH is getting down to 6.6, then you're at a disadvantage with most plants. You may be a candidate for using crushed coral to raise the hardness and eventually the pH of your tank very gently and steadily. Despite things you'll hear, most plants do not do as well in soft, acid water. About the only thing that does well in that type water are a few types of fish that get their breeding stimulated by it. Not saying to run out and do this right away, just giving you ideas to learn about.
Another thing that I believe frequently happens to beginners or anyone with new tank setups is that the nutrition of the plants is not enough. Plants need rather a lot of nutrients and they need to come into the tank in the right amounts at the right times. A common technique for your type of situation would be what's called "reduced EI" (EI stands for Estimative Index and the planted tank section pinned topics go over it) which is basically a way to add fertilizers (ferts are divided into macro-nutrients and micro-nutrients) and then take out the "excess" fertilizer at the end of the week with the weekly water change so that you "reset" the amounts.
One plant nutrient, carbon, is special. Its extremely important because plenty of it is needed by the plant to make glucose, which transports energy around the plant for all the cells to "eat," so in a sense its the "main meal" of the plant. Carbon is also notoriously difficult to get to the plant because in its ideal forms you can just pour it into the tank like other nutrients. Thus the whole planted tank skill area of CO2, which is another topic not to get overwhelmed by but to maybe start reading about. Now, recently, there are new things on the market called "liquid carbons" (such as Seachem Excel and another product, EasyCarbo, I believe.) These do in fact make it possible to "pour in carbon" but unfortunately they do not necessarily perform as well as CO2 and they act differently from it. Despite this, these often work well with low-light but are expensive.. but they might work for you at some point.
Finally, we get to algae. Its pretty darn hard not to have brown(sometimes reddish) algae (diatoms?) in a new tank. The little pockets of ammonia are just too common that trigger this algae and things are just still too unbalanced by the new aquarist. Algae is a "skill set" and to understand it often takes already understanding the skills of light, nutrition and carbon first.
Now! Hopefully some other members will come along and give you some one-liners so that the task seems easier!
~~waterdrop~~
