There are two ways to get from Dushanbe to Ishkashim: entirely by road, or a combination plane/road route. Both routes have their pros and cons.
The road route is a brutal two-stage run, first from Dushanbe to Khorog, and then Khorog to Ishkashim. The first stage is at least 17 hours; guidebooks and others say it averages 22 hours. If you leave Dushanbe around 7am (from the car park near the airport), you can be in Khorog by around midnight. Shared taxis/vans will do the first stage in one long day, which is brutally uncomfortable (unless you score the front seat), though if you hire your own car/driver you can break the journey into two days (stopping in Kalaikhum). Along the route, you will pass the Rogun dam, many small villages, and a high pass where the temperature drops considerably (and gets closed in the winter months due to snow). Most of the ride, though, is along one of most bone-jarring bumpy roads you will ever ride, preventing your car from ever breaking the 20mph barrier.
From Kalaikhum to Khorog, the route follows the river/border separating Tajikistan and Afghanistan, offering amazing views of the villages on the other side (if you travel the road during the day). You will not see this from the plane. Be warned that the road may still have landmines along it. So, do not venture far from your car if you stop to take a break.
There is an alternate route between Dushanbe and Kalaikhum that follows more of the Afghan border, though it takes longer.
Once in Khorog, you will need to spend the night before doing stage two to Ishkashim. Shared taxis/vans for this stage leave from the car park across the river from the main bazaar in Khorog. See my description of this route below, after I discuss the plane option. In all, it will take you a minimum of two days traveling to reach Ishkashim with this option. In my case, it took three days, as the day after my Dushanbe-Khorog drive was the day the Afghan border was closed (Sundays)...so I was stuck in Khorog for a day.
The plane/road route is (obviously) the quickest, and it is possible to make it to Ishkashim in less than a day from Dushanbe. However, it is also the hardest to make happen. Each morning there is a single flight from Dushanbe to the Tajikistan town of Khorog. The flight is only about an hour, and leaves at 8am each day. Tickets go on sale at 7am the day of the flight only, and the flight happens only if the weather is good and enough people want to fly that day. If the flight the previous day was cancelled, ticket holders from that flight will already be ahead of you for the flight. I did not make the flight, as the flight the day before had been cancelled, and the weather was not ideal. Based on other travelers' experiences, the flight is spectacular, and gets you up close and personal with the peaks of the mountains. There are YouTube videos of the flight.
Once you land in Khorog, you can head straight for the border (2-3 hours away), or spend the night in Khorog. If you land early enough (mid-morning or so), head to the shared taxi area across the river from the main bazaar in Khorog (maybe 8 somonis from the airport to the bazaar). Here you can find a shared car/van to the Ishkashim border for 25 somonis (~$6 USD) per seat. If you land after 11am or so, it might be hard to find a shared car going to Ishkashim (most leave after the people do their morning shopping in the bazaar and head back to Ishkashim). If that is the case, you will need to rent an entire car (~$100 USD). The drive from Khorog to the Ishkashim border post is about 2.5 hours long with good weather, road conditions, and quick driver. The road follows the border between Tajikistan and Afghanistan, offering excellent views of Afghani villages and rapids of the river separating the two countries.
Check with the Afghan Consulate in Khorog or your tour company regarding the hours of the Afghan border. They are limited and it is closed on Sundays.