It's really sweet of you to be helping your mom, and sharing a hobby. The fish store will usually do free water tests for you if you bring in a sample of your water.
You can try doing a 40% water change, and adding a new aquarium supplement like "Nutrafin Cycle" that helps by giving you some of that biological filtration that MABJ was referring to in the tank cycle. Then do another 20% water change every day for the next couple of days. Follow the directions on the Cycle for how much to add.
I'd also recommend returning at the very least the black tetra. The fish store may take the black tetra back in exchange for a harlequin although they may not. Schooling fish like the harlequins would prefer to be in groups, three at the very minimum although some people would say even more is better. I wouldn't do more than those 3 in your 3g.