From Rome you can take the train to Naples and switch in the station to the local Circumvesuviana to Sorrento. Sorrento is a pretty town on the Gulf of Naples and has good public transportation to Pompeii, the Amalfi coast, and the isle of Capri. Or, for a more relaxed (but probably more expensive) visit, stay in Positano on the Amalfi coast. And make it 4 days for more leisure time.

Recommend May with 6 nights in Rome and 3 nights in Positano. I would not waste an evening in Rome by staying at a hotel near the airport for the last night. A taxi or hotel arranged driver will get you to the airport in plenty of time even if you are staying in the ancient part of Rome.
Have a great vacation.

I've been there in March and May.
May gives you longer days and nice outdoor eating.
We always spend at least a week in Rome. Your schedule looks good.