How should I travel from Sheki, in Azerbaijan, to Tbilisi, in Georgia?
Are there direct connections? If not, where should I expect stopovers?
Can I only travel by Bus? Or also by train?
How long will it take?
Well, I did the trip today. Here goes:
Total time traveled, about 6.5 hours (with the one hour time difference).
Alternatives are worse. Using the more regular Azerbaijan to Georgia crossing requires you to get down to Yevlakh, on the Baku - Tbilisi road and rail. With only one train a day doing that trip, and many more kilometers to be made, this is not very convenient.
(Balakan is the border town, about 12km from the crossing with Georgia.)
This strongly implies there is no direct bus - but is not a definitive answer either.
Another bit of information lacking is whether the train to Balakan continues on to Tbilisi or whether you would at least have to change trains. But since we know there is definitely a train between Baku and Tbilisi I would expect you can take the one train the whole way, or at least there would be a standard connection for the rest of the way since there seems to be only one train per day.
I'm going to update the above info with costs + time breakdowns.
Note: Find the bus to Balakan--it will be clearly marked--get on, and pay the driver at the end. It filled up. Try to get on a little early.
Note: They will almost certainly try to rip you off. The guy told me 6 and charged me 8 at the border after I gave him a 10.
Note: Make sure you have your Azerbaijan visa with you. They asked for it on two occasions.
Exchange $$ or Manat for Lari. I don't recall seeing ATMs, so keep some extras.
Minibus from border to Lagodekhi. All day/frequent. 2 Lari. 20-30 minutes.
Note: You can find the buses pretty easily. They are very easy and certainly cost less than the taxi option.
Note: The last stop is right next to a metro station called Grmagele.