Gille:
I use Zinsser's Bullseye 123 for most of my priming needs, but if I were you, I wouldn't use it for spindles on a bannister (which is what I presume you're planning to do).
You see, being a PVA primer, the Zinsser's is going to dry to a soft film. However, this is a bannister on a stair case, and if it were me, I'd probably be wanting to use an oil (alkyd) paint on them. (Simply cuz bannisters on a stair case are more likely to be bumped and bruised moving stuff up the stairs and down the stairs than a ceiling or wall would, and oil based paints provide for a harder film which will stand up better to both the bumping and bruising and the cleaning up of scuffs, etc.)
So, you'd end up with a soft latex primer with a hard oil based paint on top, and it's never a good idea to put a hard coating over a softer one cuz that's a situation that's likely to see the hard top coating getting chipped up easily as the one underneath it breaks and fails in daily use.
If it were me, I would probably be most inclined to use an ordinary INTERIOR oil based primer over the stained wood you have, and just allow that coat of primer to dry for an extra day or two before topcoating with an oil based paint. You MAY get some bleeding of the old stain into your primer, but you also may be surprised at how little of that occurs. There is a product called KILZ sealer which is an alkyd primer that uses a lot of naptha instead of mineral spirits as it's solvent that would minimize the bleed through of stains because it dries so fast (naptha evaporates about 5 times as fast as mineral spirits), and it would be OK to use if you ROLLED it on, but it's a bear cat to paint on with a brush. It dries so fast that you're continuously fighting trying to smooth out brush strokes, and the only effective way of doing that is to thin the primer with mineral spirits to increase it's drying time, but that effectively changes the product back into an ordinary interior alkyd primer, so you're left wondering if it wouldn't be better just to use an interior alkyd primer and rely on allowing longer drying time for better encapsulation to deal with any bleed through.
If you can, see if you can still get a true BOILED LINSEED OIL based primer in your area. Nowadays, alkyds paints are the norm, and an alkyd resin is nothing more than a "clump" of drying oil molecules (or parts thereof) and the considerably larger size of alkyd resins greatly reduces their ability to penetrate deeply into the wood to obtain the excellent adhesion to the wood that the true drying oil primers and paints were able to achieve. Basically, alkyd resins penetrate into wood better than latex resins (which don't penetrate at all), but for really good penetration and adhesion to the wood, you can't beat a true drying oil based product.
Also, whenever using any oil based products, it's best to ensure you apply each coat within 24 hours of the previous coat to get good crosslinking between coats. That, in turn, helps ensure excellent adhesion of each coat to the previous one. If you need to sand out any bumps or lumps in the paint, then you need not observe that 24 hour timing as the roughening of the substrate will ensure good coat-to-coat bonding.
But, your general gameplan wasn't too bad. If you had a glossy surface, using Bullseye 123 to stick to it, and then top coat over that Bullseye 123 makes sense at first blush. If you were planning to use a latex paint over top of the Bullseye, it would still be an OK plan. But, in my view, you'd get a more durable paint job and better performance if you went with an oil based primer and paint rather than latex on your bannisters.
Certainly, on any working surfaces, such as shelves, window sills, floors, desk tops, or anything where things get slid over a painted surface, stick with the harder drying oil based products for best results.