I would like to spend a few days in East Turkey before I go to Georgia, 3-4 atmost. Let me know if this sounds good:
DAy 1:fly into Trabzon, reach by 10 am or so. Go to Sumela monastery and return by evening. Take a night bus to Erzurum.
DAy 2: Reach Erzurum in the morning. Check out all sites by evening, leave for Kars by an evening bus(is there one?) Is morning 7-afternoon 4 PM good enough for Erzurum?
Day 3: Go to Ani and back.
Day 4: Proceed to Georgia via the Posof brder(is that possible)?

You might struggle to get from Kars to Georgia in a day. When I made the trip, there were only 1 or 2 minibuses a day from Kars to Posof, and the day I traveled, the only minibus left in the afternoon. I didn't arrive in Posof until after dark, and could only continue into Georgia the next day (I'm fairly sure the border here closed at night, at least when I made the trip).
If you're really rushed for time, why not just fly into Kars? Kars itself has enough to keep you busy for a day, in addition to Ani.

Thanks! Were you able to atleast reach Akhaltsikhe by evening? I'm planning to stay there and visit Vardzia the next day.
Um, did you read my post? I arrived in Posof after dark, didn't attempt to get to the border that day (since it may well have been closed already, and it was after dark), spent the night in Posof, and then traveled onto Akhaltsikhe the next morning. There is no local public transport between Posof and Vale (a town on the Georgian side of the border), which makes the trip a little inconvenient. You might get lucky, and be able to do Kars-Akhaltsikhe in a day, but you need a back-up plan in case there is no morning transport from Kars to Posof (or you need to hear from someone else on here that there is now definitely a morning minibus service).
About Vardzia: it is possible to visit as a day-trip from Akhaltsikhe by public transport, but it's quite rushed this way, and will be a long day. If possible, you're best off trying to spend the night around Vardzia.

Oops,sorry,I thought Posof was in Georgia. It certainly looks complicated. I am thinking of Trabzon-Batumi and then onto Tbilisi, is that possible in a day(since I am going to Trabzon anyway).
An alternative would be Kars-Hopa-Batumi,is that possible in a day?
It should be less complicated if you cross along the coast. You could fly into Kars, work your way from there to the coast in 4 days, and then cross into Georgia. Batumi-Tbilisi is certainly possible in a day, and there's probably an overnight train. I've never crossed the border at the coast, and so don't really know what transport connections there are like. If you are definitely flying into Trabzon, getting to Kars and back in the time you have will be very rushed.

Thanks! So I'll find out if Kars-Batumi is possible, if it is, I'll go for it,since I believe it would be more scenic.
Nice to see you are planning a return visit Ashwin!
From Erzurum to Kars there are only early morning buses;04.00,04.30,05.00 and 06.00 for this 3 hour journey and with Has Bingöl,Cesur Bingöl,İğdirli Turizm,Star Diyarbakır and Öz Diyarbakır bus companies.So you would really need to plan to spend a (short)night in Erzurum.
AFAIK there are no direct buses from Kars over into Georgia;you need to take a bus to Hopa and change there.
So the next thing to think about is...what about flying first to Kars,then bus to Erzurum and on to Trabzon and then bus from there.This would be easier as there are several regular buses every day from Trabzon to Georgia and that is the part of thistrip that is causing you the most trouble isn't it?

I am very excited about this! Plus this is happening all of a sudden.
I would ideally like to do a "scenic" road somewhere in this region, which one could you recommend?
I like the idea of Kars-Hopa, as long as there's a daily bus.
Is the Trabzon-Batumi road beautiful? Where could I see something like this: https://www.flickr.com/photos/93669224@N07/15339642626