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Tbilisi is a beautiful city with European-looking streets filled with signs that have the unique Georgian alphabet on them. The city is very conducive to walking, though taxis are cheap if need be. I would recommend staying in the center of everything at the Kalasi Hotel. For about 80 USD a night, you will get a clean and spacious room (you may even get lucky enough to get one with floor to ceiling windows overlooking the streets), breakfast and an awesome location in the heart of everything.
If you're planning on exploring the entire country and don't want to spend a ton of time in Tbilisi, I'd recommend at least visiting these three main attractions:
Tbilisi Funicular: The railway was built in the 19th century to connect 'Upper Tbilisi' (on the mountain) with 'Lower Tbilisi'. The city had planned to build a district up top because the Mtkvari River made it difficult to move within the city limits down below. Unfortunately 'Upper Tbilisi' was never built due to water supply issues, but the funicular remains there and offers tourists beautiful views of the entire city. Historical fun fact: The Belgian owner of the railway offered a large sum of money to the first people who would take the funicular up and down the mountain. His efforts were a success and residents lined up to try the city's new form of transportation.
Holy Trinity Cathedral of Tbilisi: Also known as Sameba, the Holy Trinity Cathedral of Tbilisi is the primary Orthodox Cathedral in Georgia. The cathedral was constructed between 1995 and 2004. It's hard to miss as it stands tall in the middle of the historic neighborhood of Avlabari. When stepping into the gates of the cathedral's complex, you will see the main cathedral, a bell tower, the Patriarch's home, a theological academy, a monastery and a clerical seminary. I'm telling you...it's massive!
Cathedral of Saint George: The Cathedral of Saint George is one of two Armenian churches in Tbilisi that is still fully functioning. It is a 13th century Armenian church located in the historical area of Tbilisi. Though the place looks grandiose from the outside, it's actually quite small inside. Females should bring a scarf to cover their heads, though most all of the cathedrals have extras in case you forget.
**Plan to road trip around Georgia with a must-see stop in Kazbegi.