2001 Honda Civic overheating

I have a 2001 Honda Civic that is having some overheating issues. Basically, when the car idles, the engine shudders at low RPM's and starts to overheat, but once you get back up to speed, the engine cools back down to normal temps. So, self diagnosing I have noticed that while the engine idles, I can pretty much pinch together both the upper and lower radiator hoses and I really don't feel any water moving through. The thermostat was recently replaced, but did not fix the issue. The water pump's weeping hole is not dripping water and I have not noticed any puddles of noticeable size underneath the car. I sort of suspect either the water pump or the timing belt but since both of those are outside of my comfort zone a little, I wanted to get some input from an expert and see if there is anything else I can check before I carry it to a mechanic and give away my first born child.

Please help!