Mtskheta (მცხეთა) is an ancient town of extraordinary importance to the Georgian nation, located about 20km out of Tbilisi. It was the capital of the eastern Georgian kingdom of Iberia from the 3rd century BC to the 5th century AD. Mtskheta was the site of Georgia's adoption of Christianity in 334 and remains the headquarters of the Georgian Orthodox Church. Mtskheta is of primary interest to any travelers interested in Georgian history or Orthodox Christianity.
You can take a taxi from Tbilisi or book a guided tour.
Marshrutkas run regularly every day of the week between Tbilisi's Didube market and the main street in Mtskheta. From the Didube metro station, you walk out and around the one-story shops that are ahead and to the right. In that lot, there is a 'cash desk' where you can buy a ticket for the marshrutka. The cost to Mtskheta is 1 GEL per person. You will need to pay attention for when you're near the historic center of Mtskheta because the marshrutkas continue to the Mtskheta bus station which is a few kilometers north of Mtskheta's cathedral. Note that there are fewer marshrutka drivers operating on Sundays and that this mode of transportation therefore becomes less dependable and more crowded.
~20GEL from Tbilisi to Swetitskhoveli Cathedral. You have to negotiate the tariff before starting.
Mtskheta is small enough that it is possible to see the entire town on foot. The one exception to this rule is Jvari monastery, which is 5 km from the town center and located on top of a mountain. As there is no public transport to Jvari, it may be best to hire a cab in town. The cab will take the (longer) tarmac road. There is a tourist office opposite the cathedral where you can get a map of the city and the surroundings.
There are some traders and a small shop in front of the Svetitskhoveli Cathedral where you can buy souvenirs and drinks.
The Georgian cuisine available is outstanding and costs far less than a comparable meal in Tbilisi. It's almost worth it to come from Tbilisi to Mtskheta just for dinner here! Another extremely popular place is "Salobie" on the way to Mtzkheta to Tbilisi. It's a large restaurant with both outdoor and indoor seating areas and the food is exceptionally good.
There is also a relatively large grocery store ("უნივერმაგ" or "Универмаг") located north of Samtavro on the main road (between Samtavro and Bebris Tsikhe) where you can buy cheap basic foods and drinks.
Mtskheta is an easy and exceptionally rewarding place to visit. The accomodation here is cheaper and generally better than the budget accomadation on offer in Tbilisi, and the easy mini bus connection to Didube metro station, makes it a good alternative place to stay.
Again, taxis or the marshrutka are the main options. Taxis are a bit more convenient, albeit occasionally difficult to find within Mtskheta; the marshrutka is cramped, but cheap. To be on a safer side get a phone number of a cab company while still in Tbilisi, they will always send a car to pick you up on the way back.