Mazda Milenia

To check water pump:

While your engine is at operating temp, (temp gauge half-way) squeeze upper hose at the same time as someone revs the motor up and down. If you feel pressure increasing as rpms rise, then pump is good. If not, pump is bad.

To check the thermostat:

Feel both bottom and top hoses. If one is hot and the other not, then your thermostat is stuck closed or there's a blockage in the radiator.

To check for a blockage in radiator:

Let engine cool down and place heater controls on high. Then remove upper and lower radiator hoses, Place garden hose in radiator and turn on water. If water flows freely, radiator is good.

Next run water through engine. If water doesn't run through, thermostat is stuck.

Finally, a blown head gasket can cause this same issue. Do you have any smoke coming out of the tail pipe at cold start-up? Is there any in the engine oil?

To quickly check (not fools proof), take off radiator/resevoir cap. While someone revs motor up to at least 1500 rpms, look at coolant. If there's large bubbles, that means you have a blown head gasket.