Markha Valley Main Trek
Starting from the new bridge 4km south of Chilling, or alternatively at the nearby trailhead village of Skiu, reasonably fit hikers should budget on four days to reach Shang Sumdo. This involves walking roughly seven-hour days and sleeping at Sara or Nakdeng, Hankar and Nimaling. If you'd prefer to do the trek in shorter sections, take five or six days, stopping instead at Skiu or Kaya, Markha, Umlung or Hankar, and Nimaling plus, perhaps, Chokdo to break up the long descent on the final day.
From Chilling and Kaya as far as Hankar you'll find villages every two or three hours' walk; these contrast very photogenically with the stark, dry, spikily upturned strata all around them. Human-made highlights include tiny Tacha Gompa, lying 25 minutes before Umlung (Umblung), and the shattered fortress ruins at Upper Hankar. Both are perched improbably atop razor-sharp ridges.
As the first two days have only gentle inclines, the Markha Trek helps you acclimatise as you go. Nevertheless, be sure not to start too soon after arrival in Ladakh.
Trekking from Zingchen
If you're well pre-acclimatised, you could add two or three beautiful extra days to the main Markha Valley trek by starting instead from the end of the paved road past Zingchen. From there it's three hours' walk to the single homestay at Yurutse, which has views of Stok Kangri; should that be full you might have to backtrack to charming Rumbak, which has nine homestays.
Start early the next morning; cross the Ganda La (4920m), two hours from Yurutse, and descend to traditional Shingo (six to seven hours), where there are three homestays. If you have the energy, you could continue to Skiu on the Markha Valley main trail in around another three hours, but unlike Shingo, Skiu–Kaya is very spread out, meaning you might have a long further walk to find that night's homestay.