Ashgabat is a city in Turkmenistan.
North towards Darvazar, Konye-Urgench, Dashoguz
From Tekke Bazaar take the bus to Gundogar Bazaar, the new Tolkuchka Bazaar (1 Manat in April 2012). Get off when the bus turns right from the main road to enter Gundogar Bazaar. Walk 200m to the monumental marble gate marking the entrance of Ashgabat and rise your thumb.
South towards Iran
From Tekke Bazaar take bus 16 (0,20 Manat in April 2012). Get off when the bus turns right from the main road a few kilometers South of Ashgabat. From there, you can easily get a ride for the last 5 kms to the police checkpoint. Do not try to stop a truck as they have to stop just a few hundred meters ahead in a truck terminal.
Hitch-hiking across the border is not possible. Instead you will have to take the shuttle across the 30kms of no-mans land to the border post. Foreigners are asked an astronomical 40 manats (in April 2012). Argue hard as it is obviously not the real price locals pay. Hitchhiker Olivier managed to get a free lift with the shuttle bus.
From the Turkmen border post, it is only a 50 meters walk to the Irani side, and another 2 kilometers to the Irani village of Bajgiran from where it is easy to get a lift to Quchan and further to Mashhad and Tehran.
If you are heading directly to Tehran, ask a Turkish truck driver at the border as all of them are driving to Turkey through Tehran. Or you can also try as the junction 15kms South of Quchan, as many truck drivers stop there for lunch or diner. Note that if you are heading directly to Tehran you don't have to go through Mashhad.