The vast majority of trim and doors I see are painted white or left natural wood. There is nothing wrong with either choice. White gives a fresh, neutral look, while natural wood gives a neutral but nature-inspired feeling. Don't feel like you have to use either choice. Straying a bit from the norm can be beautiful.
Neutral and Safe
Painting your doors and trim a medium taupe makes the trim stand out a bit but never too much. Most wall colors will go with it. If you prefer your walls to be a version of white, a slightly richer trim will help keep your space from looking too sterile. Think Benjamin Moore's La Paloma Gray 1551. I have used that color in the past when my client wanted white walls, and their windows were black on the inside. The gray was a nice transition from the black windows to the white walls.
Bold, yet Safe
Want some wow factor? The rich charcoal trim and door (above) are very elegant and sophisticated. Charcoal is completely neutral, allowing most any color to work with it for your walls. You don't need to stick with charcoal to stay neutral, however. Try a deep navy blue (such as Benjamin Moore's Hale Navy, HC-154) or even black. Many people don't realize navy blue is a neutral. The darker the blue is, the more neutral it becomes. Think about clothing. You could pair a kelly green sweater with navy pants, the pants being the neutral color. You can successfully use a dark, neutral trim throughout your whole house.
A Little Color!
Many time periods in history saw liberal use of color on doors and trim, such as the Colonial, Federal, and Queen Anne periods, to name a few. Colors were often chosen depending on the pigments that were available and affordable. While this much color may be more than you would like to see in the whole house, a special room such as the dining room or living room would be a real show-stopper!