Georgia gives many nationalities a long visa-free stay, which makes it easy to arrive with a vague plan and extend it. This route is the core; add days in the wine region of Kakheti if you have them.
Day 1 — Old Tbilisi
Walk up to Narikala fortress (or take the cable car from Rike Park), then down into Abanotubani, the district of brick-domed sulphur baths. Book a private room for an hour, or use the cheaper public bath. Eat khinkali (soup dumplings; hold by the top, don't use a fork) and khachapuri for dinner.
Day 2 — the new Tbilisi
Morning at the Dry Bridge flea market for Soviet-era objects. Walk Rustaveli Avenue, then cross to Fabrika, an old sewing factory turned into a hostel, bars and studios. Wine bars all over the city pour natural qvevri wine — wine fermented in clay vessels buried underground, a method on UNESCO's intangible heritage list.
Day 3 — Mtskheta
Half an hour from Tbilisi, Mtskheta was the ancient capital and is still the spiritual centre of the Georgian church. See the Svetitskhoveli Cathedral and the hilltop Jvari Monastery, both UNESCO-listed.
Days 4–5 — Kazbegi
Take a marshrutka (minibus) or shared car north along the Georgian Military Road to Stepantsminda, about three hours. Stop at the Ananuri fortress on the way if your driver will. Walk or ride up to Gergeti Trinity Church, a 14th-century church on a ridge under the 5,000-metre Mount Kazbek. Stay the night and see the mountain at sunrise, then return to Tbilisi.
The road crosses a high pass and can close after heavy snow. Go between late May and early October.
Money
Georgia uses the lari. Cards work in Tbilisi; carry cash for the mountains.