there are probably as many answers as there are painters.
Alla Prima method will be very diferent than glazing.
First imagine clearly what you are after. Rainbows can be very faint and they can be striking.
Somewhat transparent would be a more realistic way. Your post seems to me to be about realistic. I suggest glazing in layers. Finish your sky down to the horizon and expect to repaint the tree after the rainbow.
There is an interesting thread about tinting strength: http://www.wetcanvas.com/forums/show....php?t=1217942
So the sky is a clear blue and you are gonna glaze over with 10-30 glazes. The dificulty will be in loading the glaze with pigment and tint so as to be transparent and very softly blended.
DO A PRACTICE PAINTING....a study. BEFORE you start. You may want to experiment with acrylic inks.
This is a learning experience and even the not so successful attempts will give valuable lessons in blending and glazing.
The biggest challenge is not to stain the sky like you might stain lawn furniture or fine woods. A dark glaze over a light background will reveal textures and brushsrtokes.
Practice loading with a touch of opacity (white?) or opaque pgments (see that other thread).
another totaly diferent way would be to paint the sky perfectly and stiple is the rainbow. or airbrush. Airbrush and stiple are doing the same thing except the method is different. Stiple brush is the epitomy of dry brush and requires practice at getting smooth effects as does airbrush.
maybe this is to much.
Stippling with good stipple brushes may be the easiest.