Lots to explore in the Old Town! Here are some street suggestions: Jan Shardeni Street, Cotton Rows, Lado Asatiani Street, Arsena Street, Ivane Machabeli Street. Do not miss Anchiskhati Basilica, the oldest surviving church in Tbilisi, dating to the 6th century AD. In general, walk into any old-looking church in town and you probably won't be disappointed! Also don't miss the Clock Tower on Ioane Shavteli Street.
Ancient fortress looking over Mtkvari river in the heart of the Old Town.
In the Old Town, make sure to take a sulfur bath! The name of the city, Tbilisi, is derived from the word Tbili, meaning warm, thanks to the warm sulfur waters that flowed all around. The linked Spa seems accessible for tourists, but so are many others, so feel free to explore!
Famous flea market, with tons of stuff dating back even to the pre-Soviet period.
Most of the museums and galleries are located together on Rustaveli Avenue. Some recommended museums and art galleries include: Georgian National Museum: https://museum.ge/index.php?m=2&lng=eng National Gallery (Blue Gallery): https://en.wikipedia.org/wiki/Tbilisi_Art_Gallery Art Museum of Georgia: https://en.wikipedia.org/wiki/Art_Museum_of_Georgia