that wood trim molding is nothing special. (and because it's a diff species of wood from the floor and the doors, it will take up the stain color differently)

If you can, remove that old wood molding and upgrade it to a more modern trim (usually 2") along w/the 5" baseboard.

If you do both in white and leave the wood door as is, you'll get this look:

Here's the 5.3" baseboard (home depot/lowes has it)

Like here. do the 5" baseboard like shown above, and do a plain 2" molding around the doors.

if you painted the door and trim the same, you'd get this look (upgraded trim tho, not whats currently there)

if you left it as is:

I think it would look better if you upgraded everything

If you did it all in white, (all new trim) it would look like this:

wood doors w/the white trim looks nice. (as long as the trim work and the baseboards are new)

and if you're thinking of painting those doors:

I think black doors w/the oak floors looks cool.

On my own house I custom stained this oak door and 2" oak trim molding myself in black dye, and then ebony stain. (the floors are acacia, but your oak would look equally nice)

Window too. as you can see, I have that same 5" baseboard I posted above, in both of these shots.

You could try sanding your doors down to bare wood and applying a wood aniline dye, and then a stain to take away some of that red. if you like the black idea like I did, try that. easier to just paint.

This is SW Iron Ore. (I like BM Wrought Iron too) look how nice it is w/the oak floors and white trim

SW Tricorn Black

and some other black doors w/black trim and white trim w/oak floors. this looks so good

I love the black trim all the way, along w/black doors

bedroom doors w/all black trim, oak floors, soft white walls, a nice runner.

this looks so good

eggshell wrought iron on the walls and a shade darker w/semi glass on the trim. sexy.

Are you in the middle of a remodel? It may help to explain the larger context of the changes.

Have you considered stripping and refinishing the trim you have?

You can buy short lengths of red oak trim to try out the look of stains. A stain midway in color between the floor and door frame would probably look good. The door frame's trim looks like a good color already, so matching that may be an easy option. If red oak is not readily available, look at red birch: it takes stain well and has a nice grain.

or you can paint the trim(s) the same color as the walls, just a different sheen.

Walls, matte/flat or egg shell and the trim(s) satin.

Leave the doors the natural wood finish as is.