Really depends on your flight options, i.e. airports, and what kind of holiday you want. If you want to just relax on a beach and take in a bit of local culture then best to go to a less famous place & quite small rather than the touristy places like Sorrento, the lakes, near big cities etc. Saying that, if it's your first time then they aren't bad options coz you'll have no language issues and the quality of the services are generally high.For less-known places then here are my options, with the weather more reliable in the south, but it is end of Sept/early Oct so you never know:
Sardinia has some of the bluest, clearest water around: in the north there's Stintino, which isn't far from Alghero, north-east coast (Costa Smeralda - a little exclusive but not peak season so probably less expensive & busy, and the sea & beaches are beautiful) or south coast (Villasimius area or the other side of Cagliari). Then, there's Sicily around San Vito lo Capo, near Palermo, that is stunning if the weather is perfect. Also Apulia as mentioned but for the quietest and best little beaches then check out the Ionian sea south of Gallipoli rather than the Adriatic side. It's a bit more travel and adventure to get there coz there isn't an airport nearby but it's a lovely area and very quiet, and Gallipoli is interesting. Plus, there are loads of little towns nearby and Lecce, which is a must but rarely visited or mentioned by visitors. Finally, you've got Calabria and the Tropea area (flights to Lamezia Terme) which is also stunning. Check out photos on Google to get an idea.