I would paint it all, just for the modernity. But you can do just one floor, or even just one room if sufficiently isolated (e.g., a bedroom). You don't need to go all the way to white, if you're keeping the rest the current colors -- just go a shade or two lighter than the walls in the same color family. (White is fine too, but sometimes that's what's people aren't sure about.)

I would paint the cabinets and leave the trim

I wouldn't recommend the paint-everything-route .. but, in the kitchen, by adding a backsplash and changing up the wall color you could offset a lot of the oak-tones. The island could get a wrap of bead board and then paint it a contrasting color to add more interest to the space.

Beadboard ? really. All the paint in the world on the walls won't kill the oak overload