lodging

Have you searched this forum?

I would use GoogleMaps or Booking.com (on its map). to search hotels in the area you want.
Siracusa’s historical center is often considered the island of Ortigia.
For Noto look around the area near the Duomo.
In Agrigento the historical center is the area where the cathedral is and just south of it.
Pescara Vecchia (old Pescara) was mostly destroyed by the allied bombing in WW2. Pescara Vecchia is basically the corner north of the PESCARA PORTA NUOVA train station and the River Pescara to the north. Just look for the Cattedrale San Cetteo, that is where Pescara Vecchia would be, but Pescara will look very modern by the standard of Italian cities. It was largely built between the 19th and 20th century, including rebuilt after WW2.