There are a many interesting places around Tbilisi that can be easy reached in couple hours or less.
Just 15 km away is Mtskheta, believed to be one of the oldest continually inhabited cities in the world and the former capital of Georgia. One of its highlights is the Svetitskhoveli Cathedral. Jvari Monastery is also nearby, giving breathtaking views of the Caucasus. It is one of three UNESCO World Heritage sites in Georgia. Take a ride by public city bus to reach Kojori, to see its ruined medieval fortress, a great spot for a picnic and relaxed hiking. A little further away is Davit Gareja, famous for its monastery where the cloisters are caves in the rocks. There are also views out over the Azerbaijan plain and eagles can often be spotted soaring overhead. The city of Gori was the birthplace of Stalin and there is a large museum dedicated to his life, though with a conspicuous lack of criticism of his actions. Near Gori is the fascinating ancient site of Uplistsikhe, a city carved from rock with a history dating back to around the second millennium B.C. Other options for a day trips are Signaghi and Telavi in Kachetia region, perfect for a wine testing tour. Take a ride up the Georgian Military Highway to see spectacular views of the Caucasus Mountains and the beautiful Kazbegi Mountain at Stepantsminda (Kazbegi).  On route, stop at the old walled fortress of Ananuri.