The primers that adhere most successfully to slick surfaces are oil based, so use one of these if you want to avoid having your paint peel off. When you visit the store to buy paint, ask the salesperson to direct you to primers that will cover varnish. The most popular brands of this type are Zinsser and Kilz. However, all primers are designed to adhere to surfaces and to cover fully. If you choose a quality primer, as long as the label does not advise against using it on slick surfaces, it should create a painting surface that is stable and usable.
You can paint over oil-based primer with either oil-based paint or water-based paint. Both will adhere to oil-based primer well. However, you should allow the primer to dry fully before you apply paint. If you do not wait, the paint can smear and the primer itself might not attach to the varnish underneath. If you have no preference for oil-based or water-based paints, go with the oil-based paint, which generally works better with the oil-based primer.
Look for a primer with a label that mentions that it can be used on either varnish, glass or tile. All of these smooth surfaces are comparable to the qualities of a varnish finish, and a varnish that is capable of adhering to any of these should work well. Follow any warnings or directions closely.
Dry the paint well before touching it, and do not place any objects on a newly painted surface. Be especially careful when using tape to create patterns with the paint, as peeling the tape off can remove any paint that is loose.
Because varnish presents so many problems when painted over, you may want to consider sanding all of the varnish off before painting. This will allow you to apply the primer and paint to a surface which is much less slick.