Without seeing a picture, wondering if blue would suit and you can go from a deep blue/navy, to turquoise, to a soft bird's egg blue. If you search blue doors, you may see something that appeals to you and suits your home.

A red house would call for a pretty rich color. A royal blue from Maureen's idea, or deep green, or black, but also canary yellow. Purples and oranges, because of their relationship to red, may be hardest to work with. And remember... if you are not happy with the results, it is a small project and you can always repaint it again the following Saturday. Have fun with it!

What color is the trim? White? Then glossy Black

I would think a light or bright yellow, it will draw your eye to the door and lessen all the red on the house.