Welcome to the forum Netaxia. I don't want to just pile on so let me try to organize what you have been reading to make it all come together for you.
A proper cycle is done before getting any fish by adding an ammonia source and doing frequent chemical tests using a liquid type test kit. One that many of us are familiar with is the API master but there are others that are just as good. A cycle can also be done with fish already in the tank but it requires closer monitoring and frequent very large water changes to control chemical poisons like ammonia and nitrites.
Gulping at the surface is usually a sign that the fish are not getting enough oxygen but in an uncycled tank, I suspect this describes yours, it is often an indication that the nitrite levels are higher than they should be. Nitrites act on fish the way that carbon monoxide does on mammals, it prevents the fish absorbing oxygen into their blood stream by tying up the fish's hemoglobin with nitrites instead of oxygen. This means the fish can suffocate while their water contains a saturated solution of oxygen. A large water change of at least 70% can correct a high nitrite level until you get a proper test kit and the fish will breathe much easier. There is almost always enough oxygen in a filtered tank because the flow from the filter prevents any stratification of the tank's water that might otherwise lead to having only enough oxygen near the tank's water surface. By mixing in the surface layer, all of the tank water reaches a decent oxygen level that is close to the maximum possible concentration.
When you replace tank water with new water, you want to match the temperature to be the same or slightly cooler than the tank water to avoiud any temperature shock and to use water that can hold as much oxygen as the water that came out. The new tank water must be treated with a dechlorinator, often called a "water treatment" but read the label. The dechlorinator should be rated to remove chlorine and chloramine and to help deactivate the ammonia that comes from breaking the chemical bonds of chloramines. (Chloramine is a combination of chlorine and ammonia that is more stable in water than simple chlorine)
Please check the temperature of your tank water using a thermometer, not the dial on the heater. I never heat a swordtail to more than about 25C, 77F, although the neons can tolerate higher temperatures. Higher temperatures mean lower oxygen content so the problem that fish have when they are in water that is too warm is again often one of inadequate oxygen, not really temperature itself. Fish that tolerate warmer water usually have a lower need for oxygen which lets them be happy with the higher temperatures. Since the limiting factor is often oxygen content, it is better to compromise to the lower temperature where a warm water fish may not be at their best but at least they are not being oxygen starved.
In general, algae or other plant life is not a negative thing for fish although we may find it unsightly. One exception is the stuff called blue-green algae which is a green to dark green slimy looking thing that is really a cyanobacteria, not an algae. BGA, blue-green algae, is something that you would want to remove from a tank. Any other algae control can wait until your tank's filter has cycled properly and may actually help you control the ammonia levels during the cycle.
