Bog.ge

Probably the best option for a bank if you are a foreigner in Georgia. They always have someone speaking English at their branches, and they have dedicated English language customer service. Plus, the mobile banking app, from which you can do pretty much everything, is available in english.
I know I focused too much on their english language services, but in general, as a bank, BOG is doing pretty well. They have branches and ATMs almost all around Georgia.