When I first became interested in learning to paint with egg tempera I was a bit intimidated by the actual process of making the paint. Here is a simple way to make the tempera that is mixed with powder pigments to make paint:
Mix two parts dry white wine (less sugar) with one part egg yolk. That's it!
This mixture will last several days, keeping in the fridge when not in use. Egg yolk should be pierced with a knife and drained from the yolk sac; don't just mix the whole yolk into the mixture. Optionally run the finished mixture through a strainer to get out any stray yolk sac bits that will result in clumps in your paint.
Here's a simple visual:
There are other recipes for making tempera, but this is what I learned from the Prosopon school of Iconography. Other examples of recipes are just to use egg and water, egg and water with clove oil, or egg and water and vinegar. You decide.