It is more complicated than you realise, Chuck. I could explain, but it would be easier if you got a book on special relativity from the library. The time dilation effects are not simply because of the Doppler effect on light. You also have to do the maths more rigorously. The nearest problem to the one you describe is referred to as the "Twin Paradox" (it is not actually a paradox but it can seem that way until you understand Special Relativity); you can find lots of references and explanations on the internet.