Late spring and early autumn are best. April, May, September, and October give cool air for walking and the gorge is at its greenest. Wildflowers cover the rim in May; the cottonwoods along the river turn gold in October.
Summer (July, August) is hot up on the plateau but the gorge itself stays surprisingly cool, with shade from the canyon walls and the river at the bottom. Winter brings snow to Cappadocia; the valley is open but the staircase can be icy and the riverside restaurants are closed.
A whole day is the right allocation if you want to walk a serious section. Start early — by ten in the morning the tour buses arrive at the main staircase.