As odd as it may sound, a good place to start your Tbilisi adventure is Narikala Fortress. Built originally in the 4th century, but improved upon in the following centuries, and located in Old Tbilisi, it's a great place for a good view of Tbilisi. You can drive up to or walk (but honestly more like hike) up the hill Narikala is located. You can also take the new cable car, but I'm personally afraid of heights so I just walk. As soon as you get up there, definitely walk up and down the fortress and look out at the city. It's hard to express in a photo, but the view from Narikala is a great way to get an initial arial view of Tbilisi as a whole. There is also a church right next to it that is the newer version of the 13th century one that was there originally, but got destroyed in a fire. If you wish to go in you can, but if there is a service and you area woman, you may be required to wrap a scarf around you if you have shorts on, and you will definitely need a headscarf. There is always a church at a fortress because throughout Georgian history, preserving the church was the equivalent of preserving the culture (especially if you consider that the Georgia was once a part of the Ottoman Empire and often fought forces like the Persian Empire). You can walk around anywhere on the fortress, the only thing restricted is the cross and plaques (you can see they're located on the right in the header but there will be an enlarged picture in the gallery). You don't have to spend much time at Narikala, but it's a great start to seeing the city.