I changed my thermostat twice in a week on my 2008 Pontiac vibe. My car was blowing cold air and running hot but the thermostat would open up and it would go back down to normal. I would have heat for maybe three minutes and then it would be blowing out completely cold air again. I had a radiator flush and also flushed my heater core. What would be your suggestion. Possibly a new radiator or water pump?
Since you've flushed the system, we can assume the coolant is free to flow where it should. It could very well be that your water pump isn't circulating your coolant very well anymore.
First make sure the belt is driving the pump properly (make sure the belt isn't loose). If the belt looks good, a new water pump shouldn't be too expensive or too difficult to change at home. If your coolant is still in good shape (it should be new, since you've had the system flushed) you can reuse it. Just drain it in a clean container and use a paint filter to keep particulates out when you put it back in.
A new radiator is only needed if the old one leaks, is severely rusted, or is completely blocked. You would have noticed a blocked radiator during your flush.
Based on what you've said, I'd suggest looking at the heater core, it may be blocked or something may be restricting flow through it. The reason I think this is happening is that you mention that there is some initial heat, but then it turns cold. That would happen if there was minimal flow through the heater core. It would warm the air when the fan was first turned on, but then the air would cool down the core and there would be no more heat.
You can test this by turning off the fan for a while and checking to see if there is heat again when you turn it back on.
If the problem is air in the system, one way to get it out (given the lack of a bleed screw) would be to open the highest joint in the system and then pour coolant into the hose until it is as full as you can get it. But I'm not convince it is air in the system – if it was air that was restricting/blocking the flow I'm not sure if you would ever get heat.