April, this is a difficult thing to specify when we can't see the colours existing in your home. Why not get a range of white paint chips and tape them up where you can see them at a distance in the daytime and at night. You'll soon see what kind of white you need. You'll need an excellent primer (and several coats) to cover the red trim. It's a big job but it will look so much nicer.

Make sure you use a name brand premium paint. It's the titanium dioxide in white paint that gives you coverage, and it's an expensive ingredient. However, if you paint over and over again with a cheap paint, and still get poor coverage, it's a false bargain. My favourite paint is C2 www.c2paint.com.