Can you paint wood that has been varnished or ...

The rule is you can paint oil base over oil base or water base, and water base over water base but not water(latex) over oil. Varnish, im not sure about. Someone else might know more.

Gayle-

Either strip the varnish or lightly sand it. If you select option #2, then use an oil based primer, that accepts latex. I use one that Ferrell Calhoun makes, but I am sure that they do not have a patent on it.

If you are looking at a stain that does not have any other finish, you could go ahead and paint it.

If you use oil paint, then invest in some Penetrol. It makes the paint flow easier and produces a smoother surface. And use a white china bristle brush for oil. 

JayR 

I had the same situation 3 years ago, also in the kitchen. The cabinets were a dark walnut stain which I wanted to change to off white. I washed each component with a TSP solution to degloss and thoroughly clean the surface. Next I applied a KILZ basecoat tinted to the color of my topcoat, which I call "French Vanilla." Lastly I applied 2 coats of Kilz semigloss latex paint. New hammered pewter knobs and hinges finished off the job. 3 years later and with much use, the cabinets are still beautiful and a project I continue to be proud of.