From Tbilisi, the fastest option is the Vanilla Sky flight to Queen Tamar Airport in Mestia (35 minutes, twice daily in season). Book well in advance; the planes are small and flights cancel frequently due to mountain weather. The bus alternative from Tbilisi takes 8-10 hours total: train or marshrutka to Zugdidi, then marshrutka up the Enguri Gorge to Mestia (2.5-3 hours).
From Kutaisi (closest major city, about 3 hours by marshrutka via Zugdidi), the same connection applies. Driving from Tbilisi directly is possible but takes 8-9 hours on mixed road quality. The road from Zugdidi to Mestia was significantly improved after 2010 and is now largely paved; a standard car can manage it in summer, though 4WD is recommended for the Mestia-Ushguli road beyond.