A quick no nonesense demonstration by Scott Hutchison. Watch as he varnishes his painting called "Plastic".
 
Basic Tips:
  1. Wait for 6 months to a year before varnishing a painting. That way you can be certain that your thick layers are dry.
  2. Use a large and smooth brush that won't shed hairs while you brush on the varnish
  3. Use a modern varnish like Gamblin's Gamvar, instead of Dammar.  (Dammar dries slower, yellows over time, and contains highly toxic vapors)
  4. Pour varnish onto the painting, and brush away, looking at the piece from an angle so you can see if your piece is fully covered.
The items Scott used to varnish his paintings:
Brush:
Liquitex Freestyle 4 inch Varnish and Mural Brush
Varnish:
Gamvar Gloss Varnish