Sometimes, if your primer is white and your wall/ceiling/trim/whatever is going to have white paint over that, you may be ok with one top-coat, but you have to cover every square inch with no skipping.
Paint your walls with 2 coats of paint (in most cases): You should be ready to paint at that point.
Just spot prime with shellac-based primer (nasty, nasty). Some stains will still bleed through after some years, so you need to go nuclear. Water-stained walls? You need a stain-blocking primer. (For old plaster, I don’t recommend any paint-and-primer-in-one). Right here at the top, you will be happy to see my top primers and paint. Often plaster walls have a lot of lime so mold cannot grow in them. Plaster walls are harder to prep and paint than modern drywall, but it is great to save them, and it will probably cost you less in the end. Plaster Walls are worth saving and painting Painting Plaster Walls is not at all difficult.