Cut cardboard in in cartoon-like cloud shapes. Paint them white with slight overlay of iridescence or cover with bunched up white silk or a fluffy polyester white plush toy stuffing. Hang onstage, peeking from beneath the upper curtain or work into the backdrop.
Measure long swathes of sheer fabric in white, gray and/or black that can stretch comfortably across the length of the stage with some excess on either side. Have stagehands stationed in the wings on either side of the stage wave and flutter one or or more swathes of fabric. Work with the lighting and sound designers to create special effects that help bring these ethereal clouds to life.
Project images of clouds onto the backdrop. Work with your lighting and sound designers to create a digital film of the type of weather you want onstage. Add sound effects like birds singing, wind blowing or thunder.