You can buy a public transport card from the Bank of Georgia office at Tbilisi airport arrivals
Cost is 2 GEL.

Then you add credit for 3 GEL per day, unlimited bus or metro rides in the day.
You can’t pay for a ride on a bus or train, you have to have the card.

There are only two metro lines, one North – South and the other Centre – West.
Plenty of buses.