I think we may have an answer.
The strips you are using don't test for ammonia and that is the first thing to go up when fish are placed in an uncycled tank. Nitrite only appears once the tank has enough ammonia eating bacteria to turn the ammonia made by the fish into nitrite.
What happens is this.
Fish excrete ammonia, it is their version of urine. When there are no bacteria in an uncycled tank this ammonia builds up. It is toxic to fish, unfortunately. Gradually a species of useful bacteria grows in the tank - mainly in the filter but also on every other surface in the tank - which uses ammonia as food and turns it into nitrite. Once nitrite starts to be made that also goes up until a second species of bacteria starts to grow which uses nitrite as food and turns it into nitrate. Nitrate is only toxic at higher levels so that can build up a bit without harm. Those two words, nitrite and nitrate, are very similar and we have to be careful to talk about the right one.
The first thing you need to do straight away is a water change. Remove at least half the water and replace it with dechlorinated water warmed to roughly the same temperature as the tank water - feeling it with your hand will get it close enough. And do the same tomorrow and the day after, and the day after etc.
The next thing is to buy an ammonia tester. Liquid reagent ones with test tubes work out cheaper and you will be doing a lot of testing. If you run out of strips, get a liquid nitrite tester too (nitrite not nitrate!). When you have a tester for ammonia you will be able to determine how often you need to change the water but until you have a tester, change 50% every day. Doing these changes won't harm the fish but not changing the water if there is ammonia present will harm them.
Ammonia must be kept below 0.25 by doing water changes. Your job is to keep the ammonia level below that number by doing water changes. After a few weeks you will notice that it takes longer for the ammonia reading to go up, and soon it won't go up at all and will stay at zero. But you also need to test for nitrite at the same time. At first there will be no nitrite but as the ammonia eating bacteria start to grow they will start making nitrite. That will go up faster and faster and you will need to do water changes to keep that below 0.25 as well - even if ammonia stays low you will need to do water changes for nitrite. Then you'll realise it is taking longer for nitrite to go up and that too will eventually stay at zero. Once you have zero ammonia and zero nitrite for a week without having to do any water changes, the tank will be cycled.
However, it will take a few weeks.
After 10 days you might well have grown some ammonia eaters; enough to make a bit of nitrite. But you still need that ammonia tester to keep the fish safe.
You can also help keep the ammonia and nitrite levels slightly lower by feeding the fish every other day rather than every day.
But I have to warn you that the sickly tetra might not make it. If this does happen, don't buy another fish yet. You can get another one (or better, a total of six rather than 4) once the tank is cycled.
You don't mention the size of the tank but if it is big enough for more than just the tetras you need to add more fish a few at a time. You will have grown just enough bacteria to eat the ammonia made by the fish you have now. When there are more fish they make more ammonia so you have to grow more bacteria. Adding too many fish at one go means the bacteria would have a lot of catching up to do.
Shops in general do not tell you how to cycle a tank properly. They rarely mention fishless cycling and very few tell you what to do during fish-in cycling. As you are finding out, they often give bad advice. The first lesson of fishkeeping is don't listen to the shop. Do your own research
