No one has ever returned from Tbilisi without a photo of the view from Narikala Fortress. The cable car starts from around the Bridge of Peace area, it is actually pretty modern. When you get to the top you will be able to see the amazing view of both the city center and the statue of Mother Georgia. It symbolizes the fact that Georgians are extremely hospitable people, but if someone attacks them, they are ready to fight back. When there you should go up to the remains of an IV-century Narikala fortress (the legend has it that it was one of the stops on the Silk Road).
There is actually another cable car in the city – it is called a cable railway. The cable railway goes up to another mountain, this time to Mtatsminda Amusement Park which I highly recommend visiting even if you are not into the rides – the 360-degree views are fantastic.