. . . . . . . . sun came out right after . . . . . . . . rain and there was . . . . . . . . beautiful rainbow in . . . . . . . . sky.

In the first blank, article "the" will be used because article "the" is used with the things only one (the sun, the moon).
In the second blank, article "the" will be used because 'rain' is definite here.
In the third blank, article "a" will be used as the sentence is referring to something general.
In the fourth blank, article "the" will be used because here the sky is specifically mentioned for a specific day.