Hawaii has the most rainbows in the world. A critical factor in producing these rainbows is the mountainous character of the Islands. Although each Hawaiian island has unique topography and attendant orographic affects, in every case the mountains produce sharp gradients in clouds and rainfall that are key to abundant rainbow sightings.
Iris from Classical Mythology was a goddess that acted as a personal messenger of Hera, rainbows being her iconic method of transportation between Earth and the Heavens. She has a sister named Arke who performed a similar role for the Titans during their war with the gods, who represented the secondary faded rainbow sometimes seen in the shadow of the more prominent one.
From Norse Mythology, the Bifröst is the name of the rainbow bridge that connected Asgard (home of the gods) to Midgard (home of humans). The Midgard end could be moved around, but the Asgard end always connected to Heimdall's home of Himinbjörg. Like everything else in Norse Mythology it's destined to be destroyed in Ragnarok when the forces of Muspelheim storm the bridge to get to Asgard.
For a relatively modern legend about rainbows, let’s consider L. Frank Baum’s series of books about the land of Oz written in the early 1900’s. In the Land of Oz, Rainbow was a personified character who was created by the Rain King. Rainbow was the father of several of the fairies of the land, including Polychrome. Of course, while most modern readers aren’t as familiar with Baum’s fourteen books about the Land of Oz, you likely are familiar with the classic movie where Dorothy, played by a young Judy Garland, wistfully sings of going “somewhere over the rainbow.”
The Irish tradition of a pot of gold at the end of every rainbow is certainly not the oldest rainbow myth, but it may be the most pervasive in western culture. You’re almost certainly familiar with the particulars: a wealth of riches is stored at the end of every rainbow, which is carefully guarded by a leprechaun. But what is the origin of this legend? The details are a bit obscure, but it’s said the legend sprung from a story about a husband and wife who were farmers. A leprechaun cursed them for their greedy ways, storing their coveted treasures (you guessed it) at the end of the rainbow, which was just out of their reach.