Rob & Rose
Your varnished cabinets can be painted and look great. The key is to prep and prime the cabinets first. You will need to clean them with a detergent solution, I suggest TSP or DirtX TM.
Next, remove the hardware ( at least the pulls or knobs). Lightly sand all the surfaces with 120 grit sandpaper.
Apply a bridge primer. For varnished surfaces I suggest you use a solvent based primer such as Kilz TM or Coverstain TM but be sure to read the label and have good ventilation!
After the primer is dry lightly sand with 220 grit sandpaper and apply two coats of semi-gloss or satin finish paint in the color of your choice.
It can be a lot of work but don't skimp on the prepwork.if you do the paint film will fail to bond which will cause chipping/peeling.
Good Luck,
rmichael