Tbilisi (or Tblisi) is the capital of Georgia and the country's largest city.
The best way out of Tbilisi depends where you currently are in the city. Basically anywhere along the road north from Tbilisi will work well for hitchhiking. Bus routes are on google maps but marshrutkas are not, so if you haven't learned how to read Georgian yet, you might find it easiest to take a bus to the spot. Not far from Tbilisi, the road splits, with one direction going towards Kutaisi and Batumi and the other way towards Gudauri and Russia.
1. A great hitchhiking spot is about 200 meters after Tbilisi Mall. Look for the hitchwiki sticker on a sign post just before the gas station. There's plenty of space for a car to pull over and everybody driving past that point is going out of the city. It's quite common to encounter other hitchhikers here but even so, it's Georgia, you won't wait long. Several marshrutkas go to Tbilisi Mall and it's even possible to hitchhike there from in the city. Look for marshrutkas 44, 51, 87, 60, 116, 117, or bus 121 to Tbilisi Mall. If you're really in a hurry, taxis to the mall are 10-15 lari from the city center. You can take Marshrutka 121 for example from Didube metro station. Its very easy from all over the city.
2. Another spot is accessible by bus 14 from the center (Rustaveli metro station, among others). The bus is very slow but also very cheap (50 tetri in 2017). Get off at the stop Agmashenebeli Ave #12 (stop ID 1179), continue walking forward across the big intersection, and hitchhike anywhere. There is still some local traffic at this point but nearly everyone going out of the city also drives by here, so you won't wait long. Make sure not to miss the stop or else the bus will turn around and you'll be faced with the choice to walk quite a ways or run across the quite busy highway (well, sometimes standing in the median of the highway with no clear way to the other side is a good way to get cars to stop for you....your choice).
3. One hitchhiking point is about 2km from the Didube metro/bus station, near the statue of David the Builder on horseback. This spot is along the same road as the previous two, just closer to the city (meaning you have more taxis and city traffic, but it's very easy to get to Didube from anywhere in the city).
4. If you're going to Didube anyways, you can reduce the walking distance to 0 by taking a marshrutka towards Mtskheta and tell the driver to let you out when the S1 national road splits off from the Mtshketa road (see photo). Or take marshrutka 11, which goes all the way from Tsereteli metro station to the end of the city, near where the highway joins the Tbilisi Bypass. Get off at the last stop, by the turn to Jvari monastery. Both of these routes also pass spot #1 (above), so it may be easier to get off there if you don't have GPS and/or don't speak Georgian/Russian (just tell the driver to stop when you're passing the Tbilisi Mall, which you can't miss).
East towards the airport and Telavi
Take bus or minibus to the 'Lilo' market, from 'Samgori' metro station. From there it's quite easy to hitchhike. Also you can take airport bus from the "Station Square" (main station) and leave it before airport turn (you will see big statue and bridge). Bus turns to the right and if you missed the turn it is no problem, you will walk about 50 m. back to the road.
Second option Take bus 37 towards the Airport. The bus leaves the city and goes east on the Kakheti highway. After some time on the highway the bus will turn right onto Europe Street (the road to the Airport). At the highway intersection there is a big statue on a column on the left window of the bus. Get out at the first stop and walk back to the Kakheti Highway (about 500 metres). You can hitch there or walk for 5 mins to the traffic light and a gas station. This highway is quite busy with cars. Waiting time could be long, because of the volume of traffic, with many cars not going directly east. The further down the Highway you get/walk the better.
Southeast towards Rustavi, Merneuli, Armenia and Azerbaijan
Take metro till Isani. When you leave the metro cross the street and head right to the south. Follow this road until a big round about after maybe 10 minutes. Cross the street from the right and follow the street which leads over the river with the bride. At the traffic light you cross the the street to the left and than to your right side of the traffic, that you reach the beginning of the highway. You see on the right side the green highway sign. There after the crossroad about 100 meters it's possible to stop cars. The street is S4 and don't give the huge sign attention before the bride. You must get a car at the highway next to the river.
If you are not lucky and get an direct ride after 6 or 7 kilometers there is the first round about which spilts the way to Azerbaijan border to the left (Rustavi) and to the right (Marneuli) to the Armenian border. Both borsercrossings are walkable, you don't need a ride which crosses the borders. To Armenia it is the road S6.
Southwest towards Manglisi, Tsalka
Busses 10 and 106 should take you to Kojori, where you'll be far enough from the city to hitchhike comfortably.
Another possible starting point is Tskneti (Busses 81 & 34), where you could check out the remains of the 2015 Tskneti land slide on the way, which seem to have become somewhat of a local attraction.