Odzun sits on a plateau above Alaverdi, roughly 3 km uphill from the town centre. Without your own vehicle, the most straightforward approach is to take a marshrutka or taxi from Alaverdi, negotiate a short wait while you visit the basilica, and return the same way.
It is also possible to walk from Alaverdi: from the bus station, head south along the main road toward Vanadzor, turn right near the southern edge of town (close to the rail tracks), and follow the road uphill through several bends until you reach the village. The climb takes around 45 minutes to an hour. Taking footpaths through the gorge makes it more pleasant when the weather is not too hot.
Alaverdi is itself connected to Yerevan by marshrutka (roughly 2.5 hours) and to Tbilisi by road through the Debed valley.