Hi Maebee!!
I did a twelve night mediterranean cruise a few years ago - it was just fantastic! We didn't bother with the organised excursions either and we had no problems! In Rome for example the ship docked at Civitavechia, which is a fair distance from the city, but all we had to do was hop on the train and we were there in no time! Got all the major sites in too - on foot - Vatican, Trevi Fountain, Colosseum, Spanish Steps, Basilica of Saint Peter etc. I reckon it is much better to do your own thing as you can go at your own pace and either get more or less into your day! Definitely beats hanging around waiting for all the people in the group to return! The only possible advantage is that the groups might get to skip the queue - not sure if they do though! Get yourself a map, lather on the suncream, have a credit/cash card & some dosh in your hidden pouch and away you go - explore to your hearts content! A word of warning though - make sure you leave plenty of time for the journey back to the ship - they will set off with or without you!! My husband was a stickler for timekeeping - afraid of his life he'd be left behind in the gutter - so we were always back in plenty of time! Aim to be back maybe an hour & a half or two hours before departure so then if you have any delays you should be covered!
You're obviously fairly internet able so why not get some transport details/maps etc. on some of the sightseeing websites! The cruise should also provide details & maps too!
You'll be fine! Get up early and be ready to get off the ship when you dock!
You'll have a ball! Enjoy!
Spicey