Heater blows cold air only. Engine is at normal temp. New thermostate. Water flows though heater core.

If both hoses are hot, the problem could be the inside...could be a blend door actuator

check to make sure the blend door is changing from cold to hot. With the engine off and the key in the run position set the blower motor to the highest speed. Turn the temp dial to full cold and wait 30 seconds. Now turn the temp dial to full hot a listen carefully. You should hear a difference in sounds as the blend door sweeps through its full range.

If it is working and the engine is fully warmed up feel the heater hoses where the pass through the fire wall. One should be very hot and the other should be pretty warm. If one is hot and one is significantly cooler than either the heater core is plugged or there is air in the heater core causing a block in flow.

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Can you hear the blend door moving when you change the temp from cold to hot or vice versa? If not, then you need new blend door actuator. Also check the heater core for any blockage. (2 ways to tell....run the car til normal temp, and turn on the heat and feel the hoses by the firewall...if one is hotter than other, there is a blockage inside the heater core. Or disconnect both hoses at firewall, and put garden hose in one of those 2 open tubes and turn on the water...good flow or weak flow coming out of heater core tube? Weak flow means blockage) If there is a blockage, time to get new one.

heater control valve shot

Does the engine actually warm up enough to produce hot air thru the heater? In other words, does the radiator get hot to the touch? Does your temperature gauge on the dash indicate a normal operating temperature , or does it show cold? Make sure the linkage from the temperature selector in the dash is connected to the water control valve for the heater.

My guess would be the engine is indeed warming up, but a faulty (or disconnected) water control valve to the heater is preventing heated water from ever reaching the heater core.

may need a new thermostat or just low on coolant

check to see if both heater hoses are hot if they are then there is something wrong with the heater panel controls

Feel of the heater hoses going to the heater core,if they are not getting hot,then remove the heater hoses and flush the heater core out with a water hose.Also be sure to bleed all the air out of the cooling system.

I think you could be right about there being air trapped in the system.
Or the heater core(radiator) is plugged.
Either way you need a pressurized radiator coolant pump to put the coolant in without air being trapped.
Although,i have seen it done by jacking the vehicle up high enough that the radiator was now higher than the heater core level, and then the coolant was put in.
But this is dangerous to try.

Make 100% sure before you go and spend more $$$ that (1) you have good coolant flow through the heater core; get engine to operating temp,then with heater fan on high and temp control on full hot carefully grasp heater in and out hoses(i use a in-fared heat gun to diagnose). They should be very close to same temp, if not, core is plugged or if equipped with a flow control valve, its not opening or there is an air lock in system.(2) If coolant flow is good, suspect the hot/cold blend door being stuck in cold position assuming you have good air flow. Its very rare for engine cooling to be ok and heater core coolant flow be bad , caused by a failed water pump! Hope this helps! Please rate me,Thanks!