I'm trying to paint a white German winter uniform for the first time. Obviously just painting it white and leaving it at that doesn't even come close to being realistic. I tried making it just slightly gray, but that doesn't look right either. Suggestions?

Its better to blend in some colour to the white for the base colour. this means that adding more of the chosen colour you can easily make shadows and less for highlights ... keeping pure white for the highest highlights only.
You should choose your colour carefully. Browns tend to have high red concentration, reds are warm colours. As you are probably having a winter scene, mix some brown and blue or purple first to be your chosen colour. this will give a cold hue to your winter whites. Greys work as well. The trick is not to use too much colour ... just enough to convey depth in whatever scale you are working in.

I use oils and enamels, and have used Mark Bannermans explanations when starting out. See his "painting whites" article on Missing lLynx for some more info.

Frank's comments are excellent. I would add that troops often lived in these uniforms for months, and they got filthy, particularly knees elbows, sleeves, etc. Indeed, the Germans sometimes had to add white cotton smocks to wear over the white uniforms, because the snowsuits were no longer white enough to serve as camouflage. You can experiment with painting techniques to depict this ground-in dirt. Dry pigments or pastel chalks might work here.