Khorugh is on the Pamir Highway between Dushanbe, the Wakhan side valleys, and Murghab. Shared cars from Dushanbe usually leave very early and take a long, rough day in good conditions; delays, road closures, and overnight breaks are normal enough that you should not plan this leg tightly.
For onward travel into the Pamirs, ask around the bazaar and shared-taxi stands in the morning. Vehicles normally leave when full rather than to a fixed timetable. Flights from Dushanbe have existed at times, but mountain weather and aircraft availability make them unreliable; check locally rather than building a trip around them.