If you want to varnish your egg tempera, or work over it in oil, you must isolate it with a layer of blonde shellac or liquin. I recommend either of these rather than spray fix to isolate it best. Varnish is not required, and you can also "buff" or "polish" an egg tempera, once it is more thoroughly dry, gently with a soft cloth to enhance it's beautiful natural sheen.
Three more important recommendations:
Choose a rigid support. Choose a thick four-ply board with a smooth absorbent surface such as Strathmore cold press illustration board or Crescent hot press illustration board to practice on at first. I don't recommend paper or bristol because it will warp with the addition of too much water and the paint may crack. Egg tempera is a rigid medium, so choose a rigid non-bendable support. The most ideal surface for egg tempera is a rabbit-skin glue gessoed panel (such as poplar wood or high quality birch plywood). Panel is a different experience too, than working on rag illustration board, but in both cases it is possible to build up many layers. Acrylic gesso will not absorb the egg tempera paint as well as rabbit-skin glue gesso, and the paint may not bind as well. I've tried clayboard also, but only with mixed results, so I don't recommend it.
Apply the paint very thinly, and in layers. A common mistake most beginning egg tempera painters make is to apply the paint too thickly in one individual layer. Build your paint gradually and remember that the principle is translucency. You are building layers to create an illusion of color and this also results in more luminous color. Remember that the principle is juxtaposing translucent layers. Layering creates an illusion of a color---for example a layer of green, with a layer of vermillion on top would make a warm shadow on a face.
How much egg medium is too much or too little? If the paint is chalky and dusting off your surface you have too little medium. If it becomes tacky and sticky, you have too much medium. As you gain experience your finished painting will eventually have an even 'sheen' or satiny patina. You can help even it out at the end by gently applying a thin layer of egg medium in the areas without sheen.
For any other questions which may arise I recommend the Society of Tempera Painters website at: http://www.eggtempera.com and it's forum link.
Moving along next to my next miniature!......