Where do you put water in the water pump on a 2002 alero?

A worn water pump is the most likely. If the coolant is going down, the water is going somewhere.

The most common reasons an Oldsmobile Alero is overheating are a coolant leak (water pump, radiator, hose etc.), the radiator fan, or a failed thermostat. Coolant leak (water pump, radiator, hose etc.)

Oldsmobile Alero Overheating - RepairPal

Videos on overheating. You probably have air in the system from when you replaced the thermostat. The coolant system needs 'burping' of air.

https://www.google.com/search?q=2000+Oldsmobile+Alero+running+ho

Burping

https://www.google.com/search?q=burping+coolant+system

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Honestly it sounds like your thermostat is either bad or was put in backwards,

try adding steering fluid to the power sterring pump resrvoir

3.4 engine is a chain motor. No timing belt or scheduled maintenance on the chain.Water pump is less than two hours labor plus cost of the pump.

could be a loose belt or a water pump that is on it's last legs. J.

Removal & Installation

2.2L Engine

1. Before servicing the vehicle, refer to the precautions section.
2. Disconnect the negative battery cable.
3. Drain the engine coolant.
4. Remove or disconnect the following:
   Exhaust manifold, if equipped with n automatic transaxle Right front wheel Splash shield Water pump sprocket access plate from the timing cover
   
   
   
   Fig. Remove the water pump sprocket access plate from the timing cover -2.2L engine
   
5. Attach Water Pump Sprocket Holding Tool J 43651to the sprocket using the access plate bolts to secure the tool to the engine front cover.
   Sprocket to water pump bolts Engine block to water pump bolt Engine front cover to water pump bolt Thermostat housing to water pump feed pipe 2 water pump to engine block bolts Water pump
   
   
   
   Fig. Water pump mounting-2.2L engine
   

To install:

1. Use a threaded stud in the water pump hub to align the hub to the water pump sprocket.
   Water pump and bolts Thermostat housing to water pump feed pipe Engine front cover bolt and engine block to water pump bolt. Tighten the bolts to 15 ft. lbs. (20 Nm) 2 of the water pump sprocket to water pump bolts
   
2. Remove the threaded stud.
   Last water pump sprocket to water pump bolt and tighten the sprocket bolts to 89 inch lbs. (10 Nm).
3. Remove the tool.
   Water pump sprocket access plate to the timing cover and tighten the bolts to 89 inch lbs. (10 Nm) Splash shield Right front wheel Exhaust manifold, if removed
4. Fill the cooling system.
5. Connect the negative battery cable.

2.4L Engine

1. Before servicing the vehicle, refer to the precautions section.
2. Disconnect the negative battery cable.
3. Drain the cooling system.
4. Remove or disconnect the following:
   WARNING Do not rotate the flex coupling more than 4 degrees or damage may occur. Oxygen Sensor (O 2 S) electrical connector Exhaust manifold heat shield Coolant inlet housing bolt through the exhaust manifold Exhaust manifold brace-to-manifold bolt Manifold-to-exhaust pipe studs Coolant inlet housing-to-water pump cover bolt Exhaust pipe from the exhaust manifold by pulling it downward Coolant inlet pipe from the oil pan Brake vacuum pipe from the camshaft housing Exhaust manifold from the cylinder head Heater hose from the heater outlet pipe Timing chain cover and tensioner Water pump cover-to-engine bolts 3 water pump-to-timing chain housing nuts Water pump and cover assembly Water pump cover from the pump
   
   
   
   Fig. Exploded view of the water pump mounting-2.4L engine
   

continue...

I suggest that you visit the autozone.com website and register the vehicle. It doesn't cost anything and will give you access to an online repair manual. The steps for replacing the water pump should be available there. Hope this helped and best wishes.

Thethermostat is a real b!t@# to replace!
Its right nder the exhaust manifold, where the water pump is. on the far side of the engine, closest to the firewall.

I have a 1999 Oldsmobile Alero and my car was running rough and stalling. Also when the car was shifting, it was making a "thud" sound. My service engine light was on, so I took it to Autozone to run their diagnostic tool. It came back with code P0108 and they advised me to check my vacuum hoses as a possible solution. I looked and found that my air hose running from my throttle body to my map sensor was disconnected. I reconnected the hose and the engine ran normally and didn't make any "thud" sound when shifting. My car is back to normal. So check your hoses. If its still running rough, you may need to replace your map sensor.