There are daily coach arrivals and departures in several directions as well as easy mini bus (marshrutka) access to sites of tourist interest in the area.
A small town, Ozurgeti is easily covered on foot.
There are two internet cafes, both of them are close to the Church. Ask someone about 'iglesia' if you're unsure where the church is. 1 lari should get you an hour of internet. Internet cafes have skype, webcams and headsets.
There are plenty of mashrutkas to neighbouring villages (generally around 50 tetri to a few lari), and one heading to Batumi (5 lari). It costs 6 lari to take a marshrutka to Kutaisi (about 2 hours), and 15 to Tbilisi (6 hours), with buses leaving about every hour for both cities from around 10:00 until 17:00 daily.
To get to Mestia, you will need to go via Zugdidi. Contrary to popular belief, you don't need to go via Kutaisi! Take any Batumi or Poti bound mashrutka, and get off at Kobuleti. From Kobuleti, you'll get a mashrutka to Zugdidi :) Failing that, head to Poti from Kobuleti and catch a Zugdidi bound mashrutka from there. The last mashrutka to Mestia leaves from Zugdidi between 1 and 3pm, so bear this in mind!
Sticking your head into one of the buses and saying 'Ukatsravad, "place name",' can help, especially if you can't read Georgian.