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Hi TerryAnn, I live in the Vake area which is one of the nicer areas but you can still find inexpensive places around here. It is close enough to walk/hike to the Turtle Lake area which is one of the best places to go hiking in the Tbilisi area. It's also not all that far from the old town and museums. This is the best site for viewing properties: Be VERY careful with agents, I had a bad one called GPM Management and I think they've changed their name to try and hide their cons. I'm sure there are some good ones out there, I met a very nice Iranian woman I can put you in touch with if you'd like. When you try to contact people about their property even with agents you will often not get a response, this is because their English is not so good, not because you are a foreigner.  Cheers.

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.