The time of the year when you will be able to see the most rainbows is in the rainy season, particularly in the warmest rainy season. You may not see as many rainbows in the winter since rainbows are made from reflecting light in the water droplets.
In colder climates these water droplets will turn to snow thus decreasing the amount of rainbows visible. The best season to see rainbows would be in a warm, extremely rainy season such as monsoon season or hurricane season. If you do not live in these climates, do not worry! You can always find fun ways to create your own rainbows.