Book a sulfur bath session at Chreli Abano or one of the other bathhouses nearby before you do anything else in tbilisi. Private room costs around 20-30 GEL per hour and the experience is completely unlike anything in a standard spa - hot sulfur water, old stone rooms, optional scrub by a local kisi guy who will remove approximately one full layer of skin. Do it on your first evening, it will reset you after the flight.
Georgian breakfast at the hotel is worth waking up for but also dont miss the local bakeries for fresh shoti bread in the morning, the long oval loaves pulled from a tone oven that you eat warm with local cheese. There are several within a 3 minute walk of the hotel and a full breakfast costs almost nothing and tastes incredible.
For wine, skip the tourist-facing spots on the main drag and ask staff for a small natural wine bar in the neighborhood. Georgia is the birthplace of wine, 8000 years of it, and the amber wines made in qvevri clay pots are something you genuinely cannot try anywhere else in the world. A good local bar will open 15 different bottles for you if you show curiosity and the prices will shock you in a good way!
The dry bridge flea market runs on weekends and is a short taxi ride from the hotel - old Soviet cameras, medals, paintings, carpets, random objects from three different collapsed empires. Go in the morning before it gets picked over and budget at least 2 hours. Bargaining is expected and part of the whole thing.
If youre visiting in summer the heat in the old town can be serious by midday, tbilisi gets genuinely hot in july and august. The smart move is early morning sightseeing, a slow lunch somewhere shaded, and then back out in the late afternoon when the light is golden and the temperature drops to something human. The evening scene in this city is excellent so staying out late is its own reward.