Adiyaman became a hotspot after inscription of Arslantepe.
1) Nemrut Dag (within Adiyaman province)
2) Diyarbakir (3 hour bus drive)
3) Sanliurfa (Gobeklitepe) (1.5 hour bus drive)
4) Malatya (2 hour bus drive)
You can check the bus schedule from this website - https://www.obilet.com
However, large buses are not the most convenient transportation route between Adiyaman, Sanliurfa, Malatya and Diyarbakir. There are minibuses that departs hourly till midnight from each city to the other. I do not know whether any English language information in the internet but they are quite easy to find. You need to go the main bus station (Turkish: Adiyaman Otogarı). https://goo.gl/maps/vY9652Qm6K6nGKSG9
Then you can easily learn the schedule there. There are more than 10 daily return trips to Sanliurfa and Malatya, and a few to Diyarbakir.
These are the minibuses:https://www.google.com/maps/place/Adiyamanemail@example.com,38.3225,3a,75y,90t/data=!3m8!1e2!3m6!1sAF1QipMA073BltPKiEiUPc7idXBA5Nqr0Il1yEoAsOJo!2e10!3e12!6shttps:%2F%2Flh5.googleusercontent.com%2Fp%2FAF1QipMA073BltPKiEiUPc7idXBA5Nqr0Il1yEoAsOJo%3Dw203-h152-k-no!7i4128!8i3096!4m5!3m4!1s0x1533137b8ab4f00d:0x58da678bbb5525d7!8m2!3d37.7721748!4d38.322731
Divrigi (we did Divrigi + Kemaliye as a day trip from Adiyaman (4 hours with private car), but I believe this does not count).
Edit: There are 31 daily minibuses to Malatya from Adiyaman. Here is the link (Turkish) https://hareketvakti.gen.tr/adiyaman-malatya-minibus-saatleri/
I am quite sure there are even more minibuses to Sanliurfa. I do not know the schedule for Diyarbakir but it is adequate to go there as a day trip.