I'll start right at the beginning.
Fish excrete ammonia, it's their version of urine. But it is toxic to fish - it burns their skin and gills making it difficult for them to breathe. In an established tank, there is a colony of bacteria which eats ammonia and they turn it into nitrite. Unfortunately, nitrite is also poisonous - it binds to the blood stopping it taking up oxygen. But in an established tank, a second colony of bacteria eats this nitrite and turns it into nitrate, which is less poisonous than the other two.
There are none of these bacteria in a brand new tank, they have to grow and this takes several weeks. The process of growing these bacteria is called cycling (the name comes from the nitrogen cycle, ammonia -> nitrite -> nitrate)
You are doing what we call a fish-in cycle. The danger with this is that the fish can be harmed. Gasping at the surface is one symptom - becasue they can't get as much oxygen into their bodies they try to gulp it from the air.
OK so that's the background to what is happening in your tank. Now for the way forwards.
The first thing is to do a big water change. Remove as much water as you can, leaving just enough so the fish can swim upright. As long as the new water is the same temperature as the tank water and you remember to add water conditioner, this will not harm the fish. Do this every day until you have your own testers.
Do not believe anything a shop tells you. Most of them haven't a clue. That includes telling you that the water is fine when it is not fine.
Buy a testing kit as soon as you can. Liquid reagent ones are better than strips. You can buy master test kits but if you don't want to spend that much money all at once, get just ammonia and nitrite testers as these are the killers.
Then test the tank water every day for ammonia and nitrite. Whenever there is a reading above zero, you need to do a water change, a big enough water change to get the level down to zero.
At first there will be only ammonia in the water, and it will go back up again after a water change. Soon, nitrite will show up. The speed with which ammonia goes up will slow down, but nitrite will then start to shoot up fast. After a few weeks, ammonia will drop to zero and stay there - that's because enough bacteria have grown to eat all the ammonia made by the fish. But nitrite will still go up, then it will slow down, then drop to zero and stay there. When you've had a week of testing when both ammonia and nitrite stay at zero, that's when the tank is cycled.
There are things you can do to speed it up.
Tetra Safe Start is one of the most highly recommended bottled bacteria products. Adding that will help, though it's not an instant fix.
Seachem Prime detoxifies both ammonia and nitrite for about 24 hours. After that time they become toxic again, but it keeps the fish safe between daily water changes. Although they are detoxified, ammonia and nitrite still show up in the tests.
Get some live plants. Even just a couple of bunches of elodea left to float will help. Plants take up ammonia as fertiliser and they turn it into protein rather than nitrite.
But I have to tell you that you were very badly advised about the type of fish.
All the tetras you bought at first need to be in groups of at least 6 in much bigger tanks. In the wild they live in groups of hundreds if not thousands of their own kind and if there are only a few of them they get very stressed.
Then they sold you cherry barbs, zebra danios, rainbowfish and minnows. Again these are shoaling fish which need a group of at least 6 and a much bigger tank.
A 37 litre tank is perfect for a betta - just the betta and no other fish.
The best thing you can do is take all the fish excpet the betta back to the shop.
I know you don't want to hear all this but the advice from that shop is just terrible
