97 Camaro Z28, No heat during startup

My car has gotten to where in the mornings(35 degrees) when I heat the car up for 15 mins, the heat is only warm coming out of the vents. After the car has run for 30-40 mins it is really hot. I have had the radiator flushed, thermostat replaced and the whole heater hose/line assembly replaced because it needed it. I have been told that the only thing left to replace would be the heater core. What I don't understand is, if the car doesn't get warm in the morning when first started, then gets really hot after awhile, how can it be the heater core? Seems to me that I wouldn't get any heat at all?