Azerbaijan-Georgia by bus/taxi
If you plan to travel from Baku to Tbilisi by bus then your starting point would be bus terminal called locally 'Avtovagsal' (near '20 January' Metro Station). You can purchase tickets for buses that depart at 8, 9 and 10 p.m. But sometimes, all three departure times are for a single bus that departs at 10:30 p.m. Best if you come early in the day to buy tickets ahead of time. But if you can't get tickets at the counter, you can pay the driver directly if there are still seats on the bus. The bus ride is about 8 hours long, plus processing time at the Az-Georgia border (can take anywhere from 1 to 3 hours). Bus is old, but reasonably comfortable. It will make several stops along the way to let passengers off. The driver will let you off just outside of Tbilisi early in the morning (7 a.m.). From there, many taxi drivers can take you into Tbilisi for about 5-10 Georgian Lari (3-5 Euros).
Morning bus from Baku to Tbilisi leaves Baku at 10 am and it cost 13 AZN/Euro. Although LP says it is a 12 hour trip, it can make it in around 10 hours, including about an hour at the border.
Beware, that after you got through the border, it might turn out that only you would be going all the way to Tbilisi and the driver might decide he doesn't feel like going all the way to Tbilisi (a whopping 1 hour further from the Krasny Most border) just for you. And this happens with many tourists.
So don't get surprised when they ask you to leave the bus as soon as the last local leave. You can argue with the driver that you have a ticket to the terminal and that he has to drive you there. It is going to be in vain. Most likely they will not go any further and you will have to take a taxi. No money compensation of course.
So you can be basically left behind and have no other choice but to take a taxi. A ride to Tbilisi for the whole cab would be arounf 30 lari (15 Euro) that you can split with other fellow travellers.
So if you take a bus to Tbilisi you may need to be ready to pay for a taxi as well (or a marshrutka that might be cheaper than paying for the whole taxi).
By taxi If you decide to go by taxi then expect to pay around 50-60 AZN/EURO for 5-6 hours ride to Sheki, and from there it would be extra 4 hours ride for another 40 AZN/Euros that could be shared between travelers.