first of all welcome to our cruisechat. I hope you will find many useful answers to your questions.
I recommend the Mediterranean in September. In August there are a lot of children on board the cruise ships. In sptember the weather is fine, the sea will be smooth. In the early November the winter starts. Dark mornings, eraly nights, rain - and rough sea. On one of the big ships you will not feel it anyway. Do you go westbound (Sicily, Rome, Barcelona)or eastbound (Istanbul, Dubrovnik)?