Cycle takes up to 6 weeks, you have been misguided by them as they just want to make a sale as fast as possible.
I would not recommend buying from pets at home as thye have disease ridden fish and tanks, I have lost too many fish from them and have sworn that I will never again buy fish from them.
I would recommend buying fish from Maidenhead Aquatics as they are more experienced in fish and stocking, at least my local one does IMO.
Neon tetras, tetras in general, are soft water fish species and they require soft water and acidic pH.
Can you provide us with your pH, GH and KH?
KH is the carbonate hardness while GH is the permanent one and these minerals can be removed by reverse osmosis system,
@Retired Viking uses this system for his tetras and therefore is more experienced.
I'm also aware that Maidenhead Aquatics sells RO water for cheap, but you do have to have your own containers.
When you get the RO water from them, use half of it and half of your tap water mix so the water has still some minerals the fish could use, but there will be no residue buildup in their kidneys.
They will also require floating plants as they come from amazonian river/s wich has some shades and low light.
As for your main water parameters, ammonia and nitrite in cycled tanks should be 0ppm for both. Nitrates should be kept below 20ppm.
What is your ammonia, nitrite and nitrates?