I like the wood stairs, it add warmth to the space. I would paint the spindles white, like your trim and leave the riser and bannister as is. You could add a splash of color in a runner going down the stairs.

Short of changing out the turned spindles I would paint them black and then decide if you want the railing left in the natural wood color or paint it too.

Painting won't be cheap either as spindles are tedious to paint. Expect to pay $2-$3 per spindle because there is a lot of labor involved.

I recomment painting the banister white and maybe the handrail in a darker stain color.

I like Faireden's answer also.

Also consider painting the railing black and the diagonal stairwall trim in white as I suspect you don't fancy that much oak woodwork? (or in wall color to blend in if so desired). Leaving the spindles, treads and risers in oak. It would freshen the look and be easier to accomplish.

Your home looks very classic by the way. If you don't have a carpet runner - consider it, as it helps to quiet the noise of kids running up and down, but I kind of like that sound. :)