Stains and gloss aside, nice work!
What paint is used for painting chess?

The white pieces on the bottom picture are made with boxwood, this wood doesn't need to be stained, only polished and burnished with the same shavings made during turning. The black pieces are made with a dark wood so the procedure it's the same. All of them. the white and black pieces are varnished but this step can be overlooked, specailly the ones made with boxwood.
In your case you might use another wood for the white pieces, a lighter wood (like maple) so you can stain with a light color. For the black pieces you can use any wood you want with a dark stain.
The best way (at least the way that gives me the best results) is to use a pore filler, burnish, stain with a water based color (I use one called Nogalina that is a dye obtained from the shell of the walnut), burnish again, and finally aply several coats of shellac (you cna find yellowish shellac that can be used for the white pieces).
I never tried to stain oak.
the glossing pieces can be achieved varnishing them (the spirit based varnishes gives more good results than the water based but are more dangerous for health), polishing them with polish paste, like used in cars, with any rotating machine and a 'cloth disk' (I just invented this word because I don't know how to say it even in my language :) ) but previosuly they have to be varnished.
Another method I just discovered is to dye them with colored wax, the pieces have to be very good polished to obtain good results but only with one or two coats is enough.

In this picture the black pieces are dyed with the method I explained before, finished with shellac, you can get a glossy finish. They are not finished with polish paste. The white pieces are only burnished, not varnished nor polished with paste.
Another example, cherry wood + pore filler + nogalina + shellac.

I would suggest using Rit Fabric dyes. I used Tangerine Orange for my M92 hand guards. Beech wood was the wood that Zastava used for the hand guards.

I like how well it turned out. The finish is Tung Oil. Not sure if you are able to get Rit Dyes in Poland but I am sure that there is a brand of fabric dyes that are available in your neck of the woods.

I did not realise how much wood type matters but i will also check fabric dyes, i have time to experiment
Hello,
recently I did start making some chess pieces (I know that those are not perfect and have glitches
):
But I am unhappy with colors for two reasons. I am using "light oak" stain but it makes pieces darker. I would like to achieve color yellowish color like on this image:
Second thing is lack of gloss. What would be best way to achieve this "shine" effect?
Maybe somebody did have some experience with working on own piece set. I would be happy to hear some tips