Flights & getting there
From about mid-April to mid-October Gareji Line runs a daily minibus between Tbilisi and Davit Gareja, departing at 11am from Pushkinis skver, just off Tavisuplebis moedani, and getting back there about 7pm. You get 2½ to three hours at Lavra and Udabno monasteries, plus a stop at Oasis Club on the way back. The ticket allows you to come back any day, so you can stay over at Oasis Club if you wish.
Many hostels and agencies also run day tours from Tbilisi, or you can do the trip alone if you have your own wheels. If you're driving yourself, take the main road to Sagarejo and then the road through Udabno to the monastery.
From Sighnaghi, a taxi day trip to Davit Gareja costs around 100 GEL, or you could take a Tbilisi-bound marshrutka as far as Sagarejo and a taxi from there.