The 20-minute taxi ride from Zugdidi to Abkhazia’s southern boundary at the Enguri Bridge (Enguris Khidi) costs 10 GEL. There are also marshrutky (1 GEL, about hourly 8am to 6pm) from the small street off the south side of Rustaveli, just west of the river. You show your documents at a Georgian police post at the southern end of the 1km-long bridge, then walk across the bridge and show them again at the Abkhazian border post, where guards may need to call the Foreign Ministry in Sukhumi before letting you proceed.
From the border you might find a bus going all the way to Sukhumi (R250, two hours); a taxi should be R2000. Otherwise take a marshrutka or shared taxi (both R50, 30 minutes) or taxi (R200) to the sadly dilapidated town of Gali (Gal in Abkhazian), 15km northwest. Buses and marshrutky to Sukhumi (R200, two hours) leave Gali’s bus station approximately hourly from about 8am to 5pm.
Returning south, buses and marshrutky to Gali leave Sukhumi's bus and train station, in the northwest of town, about hourly from 6am to 5pm. The last marshrutky from Gali to the boundary can leave any time from about 4pm.