Loud fan and temperature gauge goes up

The first suspect should be the radiator. Run your fingertips over the cooling fins. If they crumble like a rotten chocolate chip cookie, they aren't able to give up the heat to the air. The clue is the heater core, which is a small radiator, is doing the cooling when you turn the heater on.

Feel the upper radiator hose when the engine is running hotter than normal. It should be too hot to hold onto for very long. If it is just warm, there is not enough circulation through the radiator and it may be partially plugged. You may be able to feel cool and hot areas across the radiator. If both the upper and lower hoses are really hot, the radiator is not giving up its heat. That has to be due to a serious butterfly collection in the fins or the fins are corroded. Or something is blocking the air flow like a piece of cardboard that someone put in there.

No hot air until accelerating is typical of low coolant level but it can also be caused by a leaking cylinder head gasket. A leaking head gasket can also cause overheating because if combustion gases sneak into the cooling system it can pool under the thermostat and make it not open. Thermostats have to be hit with hot liquid to open. Hot air wont do it. When the thermostat stays closed, it blocks coolant flow to the radiator.

Your mechanic can do a chemical test at the radiator to check for a leaking head gasket. That involves drawing air from it while the engine is running, through a glass cylinder partially-filled with a special dark blue liquid. If combustion gases are present, the liquid will turn bright yellow.

The lower hose is supposed to be much cooler indicating the heat was given up by the radiator. Be sure the coolant is full. Check the level in the radiator and the reservoir after the engine has cooled down.