Tblisi is very much centred around the river. Some nice old buildings, and some interesting signature bridges. It's all in a valley, with a couple of cable car routes to viewpoints in the surrounding mountains. It's pretty easy to find good value accommodation.
The main rail station is at one end of the town (and it's not the pretty end). The underground is fun - especially the escalators that are set to travel very quickly. Getting out of Tblisi a couple of hours will give you access to some pretty amazing mountain areas (if you head north), and wine regions (if you head east). You'll need to hire a car to realistically do this. Driving in Georgia requires you to maintain your wits about you.
Journeys within Tblisi (including to the airport) are 1GEL for each journey, including changes. As of today, it's about 20p. You can use contactless credit cards on the buses and metro, so no need to buy tickets.
At the airport, there are lots of people trying to get you to use their taxis, with various scammy methods. Stick to the bus, or use Yandex or Bolt (the two main ride-hailing apps). Local Sim cards are available at the airport with a massive range of prices, so don't just take the first one you see.
Georgia is famous for its local food and wine (and chacha, which is fiery). But be careful in Tblisi centre - there's plenty of dodgy establishments that look great but you probably want to avoid. Try to go to super local places that aren't trying to look attractive to tourists.