As rightly pointed out by you, refraction takes place and then reflection takes place in the formation of primary rainbow.

• The refraction takes place because air and the rain drop are two different mediums. Hence, light bends through as rain drop is much denser than air.
• Then, within the raindrop reflection takes place which is bouncing of the light wave as it touches the back surface of the rain drop.
• Again as the light passes from the drop to the air, refraction takes place.

Rain drop is not the same as a plane mirror. However, it can be compared to a tiny prism.  In prism also refraction takes place. Just as we get a spectrum in prism, we see rainbow colours in case of rainbow formed from a rain drop.