Cycling is growing good bacteria to remove poisonous fish waste which takes a few weeks. The best way is growing them before fish are put in the tank by adding ammonia from a bottle to simulate fish waste. But as you have fish, you can't do this, so you are now doing a fish-in cycle.
Fish excrete ammonia (it's their version of urine) and in a brand new tank like yours there are none of the good bacteria to 'eat' this ammonia so it builds up in the water where it burns their skin and gills - and it's hard for them to take up oxygen from the water with burned gills so they go to where there is most oxygen, just under the water surface.
Eventually the ammonia eaters start to grow and they turn ammonia into nitrite. This is also poisonous - it binds to the fish's blood so it can't take up oxygen (similar to what carbon monoxide does to us). In a few more weeks, enough nitrite eaters will grow to remove all this nitrite.
Because these two things are poisonous we need to remove them from the water by taking a lot of the water out of the tank and replacing it with nice new clean water.
To do this properly you need testers for ammonia and nitrite and test the water every day. Whenever the reading is more than zero, do a water change. It is important to add water conditioner to the new water and warm it to roughly the same temperature as the tank.
The previous posters have mentioned live plants. Plants use ammonia as fertiliser and they turn it into protein not nitrite. If there are a lot of plants, they can take up all the nitrite made by the fish in the tank.
Fish like floating plants and these are particularly good at taking up ammonia. Even several stems of anacharis left to float will help. It might not look pretty but the betta will like it.