I generally go up the hill at a point that on Google Maps is called "Kızılçukur Seyir Tepesi", at the end of a side road from the Ürgüp-Ortahisar road, shortly before Ortahisar to the right. I have been there as the sole guest of the people who sell drinks, souvenirs there. But in the later 2010's one can see crowds of Korean and Chinese tourists being run here in fast cars, to get a glimpse of the Rose (or Red) Valley at the foot of the hill. They take selfies, walk a small bit, then are transported to another destination.
I now know one can go up the hill, down into the Red (or Rose) valley, but also walk to Cavusin along the flank of the hill on a path that almost remains level. I here show pictures I took walking to Cavusin, and a few I took after having climbed the hill from Cavusin, and then back to where I started. As amatter of fact walking back I found there is another level path, higher than the one I started on, but lower than the path on top.