Arriving from Zerzevan South of town, I started my visit to Diyarbakir at the Ten-Eyed-Bridge. It's located a few kilometers down river from the old city and offers great initial views of the city. The walls sitting high on a hill look massive from afar and must have impressed any traveler (or army) arriving as I did from the South.
I continued my walk along the Tigris river. As mentioned by Stanislaw, you can't walk along the river, but have to follow the main road. Before entering the city at the Mardin gate, I briefly visited Hevsel Gardens. I get the idea of communal garden, but felt this would be better served as intangible heritage. There really is nothing historic to see and the local peasants were probably wondering what I was looking for.
After dropping my luggage at my hotel, I walked along most of the circumference of the walls. I was expecting being able to walk on the walls like in Lugo or Avila. This is only possible at some places. In comparison, I found both Avila as well as Lugo better preserved. And walking on the walls had me worried as there are no rails and the walls are not always in best shape. Highlight of the fortification are the Diyarbakır Fortress and the South Western corner.
The city itself offers more sites. To me the two best site were the Great Mosque and the ruin of the St. George Church. The Great Mosque itself was originally a Christian church (St Thomas). When you visit you can clearly see that this is a church nave layout.
Getting There
Being a big town, Diyarbakir has loads of bus and plane connections.
I came from Mardin via Zerzevan. Plan was to catch a dolmus, but I couldn't find the dolmus station in Mardin. It's not the otogar. The bit from Zerzevan to Diyarbakir I hitchhiked. I got off before we reached the town proper and walked to the Ten Eyed Bridge, only to get a second ride offered by another friendly Kurdish local.
I left by plane to Izmir. Cab to the airport fare was a joke.
And Clyde is right: You will pass plenty of checkpoints, so keep your passport within reach.
While You Are There
Zerzevan is a must. Mardin and Midyat will probably require staying a night. The next major town is Urfa, to me a must visit, too.