score:3
I've done this at least twice and there's a new way now since I was last there.
Basically there were two border crossings between Turkey and Georgia, but a third one has now opened too.
The only problem is how adventurous you are. Because there is only public transport across the major cross on the Black Sea.
I've crossed from Georgia to Turkey twice from Vale to Posof. It always feels like adventure. The first time I was hitchhiking solo and the second time I was hitchhiking with two Chileans I met in Tbilisi who had never hitched before. There were no other backpackers either time. The first Turkish town is not so close and there are no buses but I always got a ride pretty quickly because Turkish people are extremely hospitable and kind to travellers. This route continues on to Kars if that interests you. I got invited home by a local family for Iftar and to stay the night. They're even more hospitable during Ramadan! (-:
Just recently I learned that the border crossing at Kartsakhi is now open too so next time I would love to try entering Turkey that way.
If you're a bit hardcore but less hardcore than me you can ask around and probably hire taxis at the two remote border crossings, especially if you plan it in advance.
Upvote:2
I would not dismiss flying entirely. It is not necessarily more expensive than overland travel and you will save a lot of time that you can spend elsewhere.
Travel overland from Istanbul to Georgia and then to Armenia. Travel back from Armenia to Tbilissi. Then fly back to Istanbul. For a departure in one week from now (1st of July 2015) the prices for a one-way ticket start at ~ 70 EUR. Unless you plan to hitchhike, overland travel won't be much cheaper. You can also consider a flight from Trabzon to Istanbul. Flying from Trabzon to Istanbul is even cheaper. For the same date as above, fares start at ~ 25 EUR.
The option of traveling back via Iran is interesting too. I would then arrange the visa before leaving from home. This way you are sure that you will be able to enter Iran. Or, in the worst case, you know that it will not work for sure. However, there is an additional point to consider. The train from Teheran to Istanbul only runs once a week. This is a kind of a bottleneck and requires good planning. Either, you have to split up the trip into several legs, for which you have more frequent trains, or you have to take a bus. On the other hand, you don't need to go down to Teheran. You can travel from Armenia to Tabriz (Iran) and proceed from there to Van (Turkey).
There are interesting possibilities. It is a matter of time and planning.