Recently I had my A/C charged with coolant by a professional. As of today, it's still blowing out warm air. What could be the most obvious problem, I can fix, without calling out for another professional?

If you had your unit serviced by a professional then you should call them back since many companys offer 30 days warranty on there service work.

You could have a leak in the system allowing the refrigerant out that was just put in by the service company. You are not supposed to have to add any refrigerant to a system once it is charged properly. If you do have to add refrigerant then you have a leak.

If the outside unit is NOT running then you can check the fuses in the outside disconnect if present or check the inside circuit breaker.

If the thermostat is set to cool and the outside unit will not run after checking the breaker then I suggest calling a pro since there is not much else you can do without the proper tools and the knowledge to use then to test the unit.