I'd also recommend you Vake district. I live here, it's very close to parks, to Kustba hills, where you can do different short and longer hikes in the area. Currently, CityHall started renovating the main park in Vake, Vake park, so hopefully soon it will look nicer.
Also it is walking distance to so many grocery stores, cafes, restaurants, bars, Rustaveli Avenue, where there are some good museums to visit.
I definitely enjoy living here.