I visited Göbekli Tepe back in March 2025. As mentioned, you can take the bus from the stop outside the archeology museum. The archeology museum itself is a must-see; but you need to budget at least a couple hours for it. It's giant and covers a wide range of history in the region -- including, of course, Göbekli Tepe. Once you finish the museum, you can head outside for the bus, which leaves on the 45 of the hour (sharp, too, so don't get outside at 9:47 or you'll miss it!). The bus only takes contactless credit card or Urfa Kart, so don't bring cash. It's about 1 USD. If you're coming from the airport, the bus also stops on the hour at the Havas shuttle bus stop -- you come from the airport with the havas bus, and then wait here for the Göbekli Tepe bus. Ask the bus driver when they plan to leave to go back to the city after you arrive (google translate will come in handy here).
At Göbekli Tepe, you can easily spend an hour. There's a small museum (again, I would recommend the archeology museum over this one) that gives some context to the site and its importance. The rock carvings are easy to spot and you can circumnavigate them under the big tent. The only thing I wished for was more explanation on what I was seeing at times! I eavesdropped on a couple guided tours to better understand a few of the carvings. Without that, you have to rely on some sparse signs around the pathway. However, I think this WHS overall is really well done due to its accessibility from the city and multiple options for museums about it beforehand. MuzeKart works, otherwise the foreigner price is quite steep -- around 20 euros.
While you're in Urfa, you have to try Urfa kebab and cig kofte (typically vegetarian!). They are both dishes originating in Urfa and the best versions I have had in Turkey as someone who has eaten all over the country at this point! I can recommend the restaurant Cevahir Han in the city center.