You have a lot of destinations for your 11-night vacation. Especially if it's your first time in Europe / Italy and you have a group of six, I suggest that you slow the pace a bit. Remember that each time you go to a different location, you need to have all six of you at the departure point, ready to go, bathroom breaks taken, all gear in hand, etc. (it's just human nature that someone will have forgotten something, need to stop at a store for a bottle of water, need to go to the bathroom again, etc.)
I'll assume that your flights are booked but your lodging is not?
Make your plans based on the number of nights, not days. It's just easier. For example, you're actually in Rome Saturday-Monday nights, so you have four nights in Rome, three full days plus time after arrival on Saturday and before your plane leaves on Tuesday.
Keep in mind that upon arrival at the Rome airport, you'll need to get yourselves from the airport to the train station, which is in central Rome. You will all be tired and jet-lagged and it will take you at least 4 hours door to door. But keep in mind - you'll be in Italy!!! Note that the train station in Siena is not close to the main part of town (2 km from the train station to the Piazza del Campo, it obviously could be closer depending on your hotel location). If you're ok with walking from the station, then do that, if not, you'll have to get two cabs.
Of the places you mentioned, I would drop Lucca. I've made lengthy stays in Siena and Lucca (and Florence, Venice and Rome too), and prefer Siena, though Lucca is also lovely. You could devote that travel time and time in Lucca to any of your other destinations, depending on your group's interests.
I would also vote for at least an overnight in Venice, Venice is even more magical in the evenings after the day-trippers have left, and in the morning before they arrive. If you could manage two nights, then even better. And, if your group is really into Renaissance art, then definitely visit Florence. But if not, then consider time in Venice over Florence. Even with the crowds in Venice, it's such a special place. And you only have to walk away from Piazza San Marco and that area to start to escape hordes of tourists.
In terms of day trips, check on the length of any train trips. You don't want to end up spending most of your day on the train (don't forget to include time getting to and from the train station). For example, the train between Florence and Venice is about four hours each way. You could easily choose three destinations out of those that you listed, and not have enough time at each destination to see what you want to see - and make sure you allocate plenty of time for sitting at a cafe for a coffee, glass of wine, Spritz or whatever, and people-watching.
Enjoy your trip!
