Geghard is about 40 kilometres east of Yerevan, in the Azat River Gorge. The most convenient approach is by taxi from Yerevan, or as part of a shared-taxi or marshrutka run that combines Geghard with Garni — the two sites are 10 kilometres apart and the combination makes a logical day trip.
Public minibuses from Yerevan run as far as Goght village, roughly 3 kilometres from the monastery. From Goght you can walk, hitch a ride with other tourists, or take a local taxi for the remaining stretch. The road winds through the gorge and is straightforward enough; walking it is pleasant in good weather.
Most tour operators in Yerevan offer half-day Geghard and Garni packages — these are worth considering if you want transport sorted without hassle.