I'll get straight to the point. Go here and do everything you can. Explore every corner of Tbilisi and hike (or take a cable car) up to the Mother Georgia statue and ancient fortress and look out over the whole city. Then make sure to go out at night and take part of the vibrant energy. Eat the food, try the wine (and the cha-cha if you feel adventurous) and dance at any local place. But while Tbilisi is a great city, make sure to go to the countryside too. Don't just see the mountains, hike them, cycle through the valleys and maybe even try to summit one. Be part of the wild nature that has still somehow managed to mostly avoid the tourist industry. And lastly, get to know someone or some family there. The people are among the warmest and friendliest I have ever met and will be glad to share their culture with you. But if I could give one word of caution, just avoid the taxis. Georgian drivers make Russian motorists look tame and timid.

Overall Impressions

The mountains are incredible! (like the Alps, but bigger and without tourists)
The food and wine even put Italy to shame
People are very welcoming
You can get nearly everywhere by a Marshrutka
Every section of Tbilisi seems like it was built in a different century
Cats are everywhere and in nearly every restaurant and cafe
The nights are lively
For a country that is so small, there is so much to do
The society is very open
Alcohol and cigarettes are abundant and there is no taboo
Older people are more likely to know Russian and younger people are more likely to know English
The drivers are insane/reckless
Many roads outside the cities are unpaved or semi-paved
Stoplights are virtually non-existent