I'm trying to think about special relativity without "spoiling" it by looking up the answer; I hope someone can offer some insight - or at least tell me I'm wrong.  

Suppose I have an ordinary clock in front of me and I push it back with my hands.  The force applied to the clock causes it to retreat away from me and after the push, it will travel away with uniform velocity.  Suppose further, I can always see the clock clearly no matter how far away it is.  Since the speed of light is constant, the light coming from the clock must travel a longer distance to reach my eye as it moves away.  This would make time appear to slow down?  If, on the other hand, the clock is moving towards me, the distance the light must travel to reach my eye becomes shorter and shorter, thus time would appear to speed up?

Analyzing one moving clock from the perspective of one stationary person will be inadequate to derive special relativity from.  With just that set-up, you aren't actually using the key fact that the speed of light is the same for all observers – all you're actually using is just the fact that the speed of light is finite.  With just taking into account that the speed of light is finite, all you'll arrive at is the non-relativistic Doppler effect, which is different from time dilation.