I did some research for you. There are only two airlines flying MXP-NAP, EasyJet and Meridiana. In October, Meridiana's only flights are on Tuesdays. EasyJet has 3 flights on Oct 4, but only one of them is after 9.00 am, at 3.45 pm that lands in Naples at 17.10. You'd still have to get to to Sorrento - at least another hour. By train isn't much better. Take the train from the airport to Milan central station, then train to Naples, then either suburban train or private transfer to Sorrento. If you're lucky, you'd get the 10.07 train from MXP, and get to Naples at 3.50 p.m. and still have another hour or more to Sorrento. Driving isn't any better. viamichelin.com shows about 860 km from MXP to Sorrento and 8.5 hours driving time - optimal conditions, not including pit stops.
For a start, you should count nights, not days. Expect to use 1/2 day or more for getting from city to city, so that 3 nights gives you only 2 full days of sightseeing, with a couple hours more on each travel day. Every time you move, you have to pack, (breakfast?), check out, get to the train station with enough time to find your platform and board, then get to your new hotel, check in and drop your luggage. Then you can start sightseeing. You can only move as fast as the slowest person - whether it's someone who doesn't want to get out of bed, or someone who hogs the shower, or a slow packer, well, you get the idea. Allow time for "chores." Packing light usually makes it easier to move quickly (less weight, less bulk, and less to pack and unpack). But it means doing laundry more often. Even with enough clothes for a week, you'll need to do laundry at least once. Is your group disciplined enough to do hand laundry every other night? Do you need to plan on 2-3 hours for a laundromat or will you pay high prices for having your hotel take care of it? Packing light means taking few toiletries. Allow time to shop for things you aren't packing or may run out of. It takes longer to shop in Italy - stores don't always arrange things the way you're used to, packaging looks different and labeling is in a foreign language. Something you buy at a pharmacy at home may be sold only in supermarkets or vice versa. This is often a fun learning experience, just allow time for it.
A car can be an advantage in Tuscany (BTW Siena is a Tuscan town), and pretty much necessary for a stay in an agriturismo, which could be a great idea with kids, especially if there's a pool. Don't plan on long drives, though. That could be tough on the kids, 3 of them strapped into a smallish car's back seat isn't much fun. Also make sure the car will hold all your luggage. European cars are smaller and even those with ample seating room for 5 usually have very limited luggage space.
I would put in Venice as a must-see, especially with kids, and drop either the Amalfi Coast or the Cinque Terre . . . maybe even both. They are both difficult locations to get to. There is plenty to see and enjoy in Italy without them, and your goal should be to have an enjoyable trip, not to cover ground. If your family are outdoor, hiking, biking fans, then go to one of the lakes (Como, Gardo) for a few days.
