Tbilisi has a surprisingly vibrant bar and cafe scene, centred on Rustaveli Avenue, Aghmashenebeli Avenue (the quieter 'other' main street), and the wine bars clustered around Erekle II Street in the old town.
Georgian wine is the thing to drink — the country is one of the oldest wine-producing regions in the world, and the amber (orange) wines made by the ancient qvevri clay-pot method are unlike anything you will find elsewhere. Most wine bars stock both the conventional European-style whites and the skin-contact amber wines side by side; try both.
Craft beer has arrived in Tbilisi too, with a handful of small breweries operating in the city. Coffee culture is well established; the Georgian preference is for strong espresso and locals take their coffee seriously.