Şanlıurfa (officially, and thus signs and maps always refer to it this way) or Urfa (pre-1984 name, colloquially and locally the city is always referred in this way) is a city in Southeastern Anatolia. It lies on west-east highway D400, which is assigned European highway number E90.
Urfa is quite a big city, so getting to the exit on foot takes more than an hour from the city centre. The descriptions below take the Balıklıgöl ("Fish Pond") area in the old city—the most widely visited part of the city—as a starting point.
West towards Gaziantep
Bus #33 from the central street. Get out on bus stop "Akabe Toki".
East towards Viranşehir, Mardin (and possibly Zakho in Iraq)
From Balıklıgöl, walk towards the main highway through the city by following "Diyarbakır" and/or "Mardin" signs. After walking for about 20 minutes, you'll reach the highway. Then once along the highway, turn right towards east, still following "Diyarbakır" and "Mardin" signs. After walking a slightly uphill section and in about 45 minutes since you started at Balıklıgöl, you'll reach a large roundabout - here the road to Diyarbakır forks away so from this point on just follow "Mardin" signs. You still have a further ~20 minutes to walk to get to the outlying suburbs of the city. You may start thumbing when you find a good spot with a wide shoulder at one of the traffic lights in rows, because edge of the city is still away. It's very easy to attract a ride, though, as is in the rest of Southeastern Anatolia.
If all this walking makes you upset, dream that you are walking along the Silk Road—well, you actually are, as the highway—İpek Yolu Bulvarı, i.e. "Silk Road Boulevard"—follows the route what was once the Silk Road when through Urfa.
Or! Walk like ten minutes from Balıklıgöl to the central bus station, and take a dolmus to SIRRIN. It goes really out of the city, and you can go all the way, or get off wherever you feel like.
North towards Diyarbakir
Modern Urfa is centred around a large crossroads/roundabout in the middle of the city. From here, catch any minibus heading to the north "Otogar" (bus station), which is located about ten minutes up the main road out of the city on the right hand side. It's officially called "Şanlıurfa Sehirler Arasi Otobus Terminali". The bus will cost 1.5TL per person. From here, walk back to the main road out of the city, and you can hitchhike from just north of the big intersection there. That spot is five minutes short of the intersection linking to the main highway (E99) that goes directly to DIyarbakir.
You can take the bus 64 till the end, it is going to Urfa's University. After, you can start hitchhiking or walk a little if you want. If you plan to go to the border, you will find very easily truckers to go there, if you plan to go to Harran, when you will leave the main road, HH starts to be difficult but you will find dolmus (collectives buses) for a very cheap price. French people, careful, Harran people had some troubles with french! (Feb 2012)