I've traveled in Eastern Turkey last summer and also entered Turkey from Georgia.
In Batumi you need to take the mini bus at Tbilisi Square. (https://goo.gl/maps/igPYhibW1xP2)
You need to find the bus that goes to Sarpi (the border town). You 'll be dropped at the border office.
Go through the border control.
On the other side taxis will take you to Hopa. The first city in Turkey from where you can start your travel by bus through Turkey.
I spent two weeks traveling in Eastern Turkey finishing in Capadocia and from where I took a final bus to Istanbul.
You can see my photos of every day of this trip here: http://thdk.be/travel/eastern-turkey/
I hope this information is helpful to you.