We had the older wood doors and fakey wood trim and to transition it, we replaced the door with a 6 panel, installed new trim & baseboards with primed pine and painted it all in white. You don't need to have the doors, trim & baseboard painted in the same expresso color as your vanity, it will actually add more pop if you don't. I would suggest you consider painting it with a neutral like white or if you don't want them to show as much, paint them in the same paint color as you paint your walls.

While there are differing ways to do it, here's what my painters usually do:

Let's say that the hinged door opens into the bathroom.

In that case the bathroom side of the door, as well as the edges of the door, get painted the bathroom color.

The hallway side of the door gets painted the hallway color.

As far as the door casing and jamb:

With the stop being on the hallway side of the closed door, the stop, the hallway jamb and hallway casing get painted the hallway color.

The bathroom side of the jamb and the bathroom casing get painted the bathroom color.

Where some differ is that some people prefer to have all three sides of the stop painted the same color. Some others prefer to paint the face and the hallway edge of the stop the hallway color, and the bathroom edge of the stop the bathroom color.

If the transition from one room to another was open, like a doorless archway, say going from a foyer to a dining room, then there is usually a "major" room on one side of the arch and a minor room on the other side of the arch.

In this case the foyer would be the "major" or more heavily trafficked side. So the jamb and the foyer casing would be painted the foyer trim color, and the casing in the dining room would be painted the dining room trim color.