It's been a few years since I've been, but I remember a lot of spots along Rustaveli Avenue, which is the main drag. When I was there they were also renovating some parts of the downtown and it looked like they were going to be quite skateable (smooth ground, ledges, etc). Wish I remembered the names of these neighbourhoods. But generally it seemed like they were investing crazy money to make the city look modern. Still the best part is walking around the old neighbourhoods. They have a very specific flair.
There's loads of hiking to be done in Georgia. My friends went to the Kazbegi region in the north, it looked amazing. It might be worth going to Batumi on the Black Sea coast. Use marshrutkas to get around, even in the cities.
Their wine is kinda shitty, imho. Sweet and syrupy. But I really only had it a few times, so who knows? When I was there they didn't really have good beer. It was drinkable, but nothing to write home about. Maybe they've jumped on the whole craft beer craze since. Try chacha, their national hard liquor. It's grappa-like. Pretty good stuff.