Well, to determine this, one must first determine where all the other colors are going. Pokemon has way too many types to just go with pure colors, so there may be mixes and tints/shades in there. Let us start with the pure colors first.
Red-Fire. Undoubtedly. This has been associated with fire for ages.
Orange-Here is where things get tricky. Nothing really immediately lends itself to being orange. I've always been a proponent of putting Fighting here. It may have originated from MD, but I'm not too sure.
Yellow-Electric. Again, easily associated to it.
Green-Grass. Easy.
Blue-Water. Easy.
Purple-It seems like psychic should belong to this in some way, but poison is heavily associated with purple in pokemon as well.
White-I prefer to put Ice for this, because...
Grey-Normal. The most neutral color. There can be complications associated to this, though.
Black-Dark. Easy.
Those are the most common colors you can think of, unless you want to include Indigo, which I can't even begin to think of what to put there. Those are 9 colors and most of them have easy associations that can be made to them. That's half of the types you need right there. We're going to have to go a bit off the wall for the rest, though. You can either use tint/shades, or you can mix some of the colors. Here are some of the ones that I think can be plainly made.
Pink (tinted red) - Fairy. Seems obvious.
Brown (shaded orange) - Ground. Easy.
Light Blue - Flying. This has long been associated to the sky, so it seems to fit.
Light Grey (around 25% Black) - Steel. It needs a really light grey.
There may be others you want to try from this. I've never tried to tint or shade yellow for this, but you can. Shading yellow tends to make it look Gold. Pokemon itself tends to use light green for Bug. Play around, there's bound to be things I haven't thought of yet. I can't do it in a too intuitive way myself, since Ghost and Dragon are the hardest to place.
