Georgian is the official language in Georgia and the dominating language in all sectors of the Georgian society. It is written with the unique Georgian alphabet since the 5th century. Georgian is a Kartvelian or South Caucasian language, not being related to the neighbouring Indo-European, Turkic or Semitic languages. Georgian is also the main language in Batumi and Adjara (there is no Adjarian language).
Both English and Russian are quite widely spoken in Batumi. The knowledge of English is more widespread among young people, whereas the level of fluency in Russian is higher among middle aged and older people.
Why not try to pick up some Georgian?
Before going to Georgia it could be a good investment to learn how to read the Georgian script and even perhaps to learn some basic words. Signs and information are primarily written in Georgian (sometimes with parallel texts in English). There are many international loanwords in Georgian and they are very frequent in the modern urban landscape.
One great advantage of the Georgian writing system is that it was created for this language and therefore it is built on the principle “one sound = one letter”. This means that reading Georgian words is easy, as soon as you the letters. There are no words, like in English, where one sound is written with several letters (cf. though, should). Give it a try!