Your sentence is almost perfect! Here's a corrected version with proper grammar:
Let's show the different states of a desert: a harsh, dry one and one that blooms after the rain.
Explanation: The phrase 'that blooming after the rain' sounds incomplete because 'blooming' is a verb form that needs a helping verb. Instead, we say 'one that blooms after the rain' to describe the desert in a clear, simple way suitable for a 5-year-old.