How do you ensure wood stain doesn’t bleed when painting furniture?

[#1]

Instructions will often mention to lightly sand, indicating they don't want to break the varnish coating exposing the interior stain.

Applying a coat of Rust-oleum 1l Bulls Eye 1-2-3 Undercoat Primer Sealer Stain Blocker would block the stain. You could then apply a final coat of paint over that.

Here's a couple of articles on How to upcycle almost anything with paint and How to paint over varnished timber which I trust you'll find helpful. - MitchellMc

Primer by itself will not prevent the stain from travelling up the paint. You need a primer with a sealer and stain blocker. This gives it a non-mixing medium that the stain can’t mix with, thus preventing bleed through. Do a light sand and use the stain blocker primer that should do the trick.

Doing a second coat of such a primer is preventative but not a fixed rule. Sometimes one coat is enough. - redracer01