Visited April 2018. After one year and a half the site is open now to the public. The new visitors centre, ticket booth and a parking guard booth have been built (not without a controversy), as well as a road/path leading to the site itself which was covered with a some kind of roof. When I was there, no entry or parking fee was collected. The site is pretty small – just four circle temples one next to the other. You can walk around them and look from above – it won’t take more than one hour. Recently they started new research close by but not open to the public yet and won’t be open for a longer while. Probably you will meet Mahmut Yildiz, who “discovered” the temple 23 years ago together with his father. He was helping prof. Schmidt and now works there as a security guard. For now there is no local guide there who can explain you what you’re looking at. To better experience the site you should read Schmidt’s book and what videos available in internet (suggested by Solivagant). Or you can hire a guide from new Archaeology Museum in Sanliurfa – the visit to the museum is a must, because most artefacts found in Gobekli Tepe are displayed there; admit at least two hours to see museum exhibitions.
The site is very well signposted while coming from Sanliurfa (on Tentative list). No public transportation.