The gateway is Akhmeta, a small town in eastern Kakheti, reached by marshrutka from Tbilisi's Isani or Ortachala bus stations in around two and a half hours. From Akhmeta a second marshrutka runs the 25 km up the gorge to Duisi several times a day in about 20 minutes; a private taxi from Akhmeta is inexpensive and easy to arrange.
There are no border crossings or restricted-area procedures any more — that ended over fifteen years ago. Bring a passport for the standard checks. Most guesthouses will arrange transfers from Tbilisi or from Telavi for a flat price if you prefer not to use public transport.