What could be wrong with my air conditioner?
Besides checking the fuses/breakers, the fan, and any connections on the controller, there's not a lot of user serviceable parts on an HVAC system. This is because the refrigerant lines require specialized equipment to test for refrigerant levels, a license to buy more refrigerant, and even more equipment to remove the refrigerant so you can work on the compressor.

When the outside unit doesn't turn on, checking the refrigerant is typically the first step because there's a safety switch in the system that prevents the compressor from turning on without enough pressure in the lines. Without that, the compressor would turn on and burn itself out since the refrigerant also includes the lubricant for the compressor.

Do what you can to get the warranty coverage from the manufacturer, but realize that this is a situation that will almost certainly require a professional to come out and repair for you.