The thermostat is a valve that helps your engine cool correctly. When the engine is cold, all engine coolant flows within the engine block; the thermostat is closed in that condition. As the engine gets warm (generally at 195 deg F), the thermostat opens, and hot coolant flows to the radiator where it is cooled. As you might expect this is important in the summer to protect your engine from overheating. If your thermostat were stuck open, then the car would take a very long time to heat up. Things to check:
--Is there adequate coolant in the system? When the car is COLD, open up the radiator cap. Is the coolant right to the tippy top of the fill? If not, something is wrong. Things that could be wrong include: coolant leak, bad radiator cap. The radiator cap is very important, and in most overflow bottle systems there are two seals there. One that opens at 13 psi (which keeps the system pressurized) and another that allows for flow back from the overflow bottle when the engine cools overnight. If the flow back seal isn't tight, you will never get coolant into all the areas it needs to go (including the heater core) If you have a void during fill at the heater core, you won't get any heat inside during cold weather. The fix is to test the cap, replace as required.
--Is your engine warming up correctly? Generally the thermostat should be open after 30 minutes of driving in nearly all weather conditions. From testing I've done, I've seen cars at -20 deg F that never get to thermostat open, but I don't think those are your current weather conditions? Things that would keep the car from warming up are pretty dramatic, like a spark plug not hooked up, but I think you'd know that. The car would run terrible.
--Its possible that your engine is heating up correctly but your heater system is not working correctly. On your car coolant should flow to the heater core all the time (summer, winter, all the time) There is a small door inside the heater duct system inside the instrument panel that controls air flow thru the heater core. Its possible that the control system for that door is not working correctly. In your car that control system could be a manual cable or it could be a small electric motor.
--Your call on the water pump is a possibility. I've seen metal water pump impellers corroded to nothing. I will say if that were the case, you'd likely see an overheat condition on instrument cluster cooling warning light / temperature gauge. It's possible, but I doubt it. How many miles did you say were on this car?
You might consider purchasing a service manual for your car. Haynes or Chilton's manuals are $30 well spent. They can assist you in troubleshooting the details here...