How do you paint over varnished timber?

[#1]

Before painting over a varnished surface, it's essential to dull the sheen level of the gloss with 120-grit sandpaper and then apply a specialty primer like Dulux 1L PRECISION White Maximum Strength Adhesion Primer that is designed for difficult-to-paint surfaces. By removing the sheen through abrasion and then applying an appropriate primer, you are guaranteed that your topcoat paint will adhere adequately to the surface. 

If the varnished timber was also stained, you might like to sand the surface and instead use an undercoat designed for its stain-blocking ability, like Zinsser 1L CoverStain Primer Sealer Stain Blocker.

For your paint, I recommend a robust water-based enamel similar to Dulux Aquanamel and overcoating it with Cabots CFP for additional protection if this is to be a work surface. - MitchellMc

I'd also recommend giving them a light sand and the apply a stain blocker primer to prevent the bleeding. I’ve used the Zinsser brand one a few times before and it’s great. - prettyliving