Painting over varnished woodwork requires good surface preparation to achieve proper adhesion.
Cleaning
Years of built up Pledge, cleaners and other paint repelling contaminants will need to be removed with a good thorough cleaning. A good all-purpose cleaner that works well on a wide variety of contamination is a product called "Krud Kutter."
Krud Kutter comes in a pump spray bottle, making it easy to spray on and wipe off. I then use a Scotch Brite scouring pad to scrub and de-gloss the varnish in the same step. Use paper towel to wipe the Krud Kutter off as you go.
This will leave a clean and dull substrate. (Only use mineral spirits if you will be priming with an oil-based primer, otherwise use denatured alcohol.) When finished, hang cleaning rags contaminated with solvent outside or spread them out to dry on a non-flammable surface outside, away from your house and other outbuildings. (Solvent-saturated rags may become spontaneously combustible.)
Priming
Now that the varnished surface is clean and dull, you can actually prime it with a variety of bonding primers. I always play it safe by priming with an oil or alkyd bonding primer—just in case I have missed a spot here or there with my de-glossing or cleaning. Oil will be more forgiving than latex if your surface preparation is not absolutely perfect. However, don't do a careless preparation job, hoping that the oil primer will save the day. You may be disappointed with poor adhesion if you go that route. Do a complete and thorough preparation job, then use the oil primer and you should not have a problem with adhesion. My preferred bonding oil primers are XIM 400 white and/or ZInsser's Coverstain.

Sand lightly**
I follow up the cleaning with a light (but thorough) sanding. This will further “etch” the varnish and provide even more tooth for the primer to adhere to. Sanding will also resurface the substrate by removing some of the outer varnish (thus exposing clean, fresh and grime-free varnish).
Vacuum and solvent wipe
Next, vacuum off the sanding dust. Then follow that up with a solvent, wipe down of the surface you wish to paint using denatured alcohol or paint thinner (mineral spirits).
** Note: If your house was build before 1978 do not sand or otherwise disturb the substrate - you may want to have a EPA certified renovator / painter do the work as there may be lead present. See EPA information here to learn about lead paint cautions and concerns.
Painting and Decorating Concourse
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