Akhaltsikhe

All marshrutkas from major Georgian cities stop at Borjomi and at Khashuri bus station. Hence, if you missed your ride, it's not terribly difficult to catch a marshrutka going between Akhaltsikhe and Khashuri before dinner time.

(As of Mch. 2022 the prices quoted have risen 20-25%.)
From specific destinations:

• Tbilisi (Didube bus station) – 08:00–19:00 hourly. 3 hr. 15 lari. (updated Sept 2022)
• Kutaisi – 08:20, and 13:00. 3½ hr. 10-12 lari.
• Gori – 08:30. 2½ hr. 10 lari. (updated Apr 2022)
• Batumi (from old bus station near Batumi train station, via Khashuri) – 08:30, and ~10:30. 5½-6 hr. 20 lari.
• Borjomi – Every 30-60 min. 1 hr. 3 lari.
• Vardzia – 9:00, 13:00 and 15:00. 1½ hr. 6.50 lari. (updated Apr 2022)
• Nakalakevi/Tmogvi (near Vardzia) – ~08:10-08:20.

To specific destinations (vice-versa, if not listed above, should be similarly frequent):

• Ninotsminda – 08:15, and 15:00. 1½ hr. 7 lari. They obviously also go by Akhalkalaki.
• Gori – 08:15, and 13:00. 2½ hr. 7 lari.
• Kutaisi – 10:40, 15:00, and 18:10. 3½ hr. 17 lari. (updated Apr 2022)
• Batumi (via Khashuri) – 08:30, and 11:30. 6 hr. 20 lari.
• Batumi (via Khulo, Jun-Aug) – 10:00. 6 hr. 18 lari.
• Vardzia – 12:20 and 17:30. 1½ hr. 6.50 lari. (updated Apr 2022)
• Mirashkhani (passing through Vardzia) - 10:35 and 16:00. 1½ hr. 7 lari. (updated Apr 2022)

There are few direct marshrutkas from/to Yerevan, Armenia, or several connections from Armenia when making a stopover in Akhalkalaki. There is also a daily connection from Gyumri, which starts at 10:00 (3½ hr, 4,000 dram).

The direct road from Batumi to Akhaltsikhe is not served by marshrutkas, probably due its bad state and the mountain pass. Nevertheless, you should be able to catch a (4WD) ride on that road. But it will be bumpy and probably still take at least 4 hr.

The principal route to Akhaltsikhe runs from Georgia's main East-West highway (E60) at the spur in Khashuri leading to Borjomi.

There's also a route from Batumi via Gorzerdzi pass (4WD needed, closed in winter, scenic views).

There was a train in the past, but nowadays the nearest railway station is Borjomi with connections from Tbilisi and Khashuri.

• 🌍 Sapara Monastery (საფარის მონასტერი) (about 10-12 km outside of Akhaltsikhe up into the mountains; a round trip by taxi from Akhaltsikhe might cost 15-25 lari). The monastery was established in the tenth century, but the principal church, St. Sabas, was built sometime in the thirteenth century. Until the twentieth century, the monastery had been perfectly preserved, as its hidden location saved it from Ottoman discovery throughout the empire's three-century long control of southwestern Georgia. Alas, the Soviets found it, and abused it in the usual soulless fashion, albeit not to the same extent as many other Georgian Orthodox establishments—the frescoed walls were not whitewashed, and remain in good condition (especially following restoration). During a visit, make sure to climb up the nearby slopes towards a rocky outcropping to get lovely views over the monastery and the valleys in the distance. Also make sure not to use flash photography in the churches, unless you want to see some seriously angry monks. If you can make yourself understood, you can overnight in the monastery's chambers. 

• Vardzia — the cliff-side cave monastery, a former city, is the biggest reason to visit Akhaltsikhe, and the region more generally. A round trip by taxi from Akhaltsikhe might cost about 50-60 lari (or marshrutka 7 lari one way, as of July 2022).
• Abastumani — an interesting and off-the-touristic-map, former Soviet spa town with still functioning sulphur and thermal baths.
• Akhalkalaki — the center of Armenian culture in Georgia is nearby to the southwest, easily accessible via marshrutka from Akhaltsikhe.
• Borjomi — every Soviet traveller’s favourite Georgian retreat, for its parks, Romanov palace, and internationally renowned natural mineral water.
• Khashuri — an interesting small town with an abundance of hammocks.

• Armenia — There are two marshrutkas to Yerevan, Armenia, at 04:00 and 07:00 (7 hr, 20-35 lari). If you miss them, go to Akhalkalaki, from where marshrutkas leave to Yerevan almost every hour. Either the same or an additional one also goes to Gyumri at 07:00 (4 hr, 20 lari).
• Turkey — You can buy bus tickets to Turkey from the central bus station. Ticket to Posof (city on the Turkish side) is US$10 and leaves at 14:30 (could not be confirm in May 2019). There's a bus to Kars with Kars Vipturizm 4449188. It goes Wednesday and Sunday at 13:00 and costs 70 TL (or 35 lari) (May 2019). Hitching to Kars is also a very easy option and can be done in far less than a day. From the central bus station, ask for the road to the Turkish border (Turetskoĭ granitsy) and start flagging cars about 1 km from the bus station down the road. The road will pass through the small town of Vale. You are able to walk across the border.