You start with a good map, some guidebooks, and the internet, including Fodors - read the thousands of posts about traveling in Italy here. If you want to see countryside, you'll need a car, but you do NOT want a car in the cities.

Here's a link to my trip report from our first trip to Italy when our kids were 9 and 12. http://www.fodors.com/community/europe/trip-report-family-of-4-enjoys-la-dolce-vita-for-2-weeks.cfm

We had a few more days, but with 11 I would probably do 3 Venice, 3 Florence and 5 Rome. I think doing the big 3 - Rome, Venice, Florence with a daytrip to the countryside is a great first trip to Italy. We've been back to Italy twice since then.

I agree with StCirq's recommendations to get a guidebook and start reading there and through the internet, particularly Fodors trip reports. Lots of good suggestions, and I think your plan will start to formulate on what you think you're family would like to do.

You have to decide what you want to see/do in Rome, Venice and Florence to decide how much time to spend in each. (We can't know what will interest you most.) Understand you lose nearly half a day moving from place to place. If there's any time left after you've made those decisions, consider any possible day trips from each city into the countryside using public trans or local tour companies. For example, while in Florence, you could use trains to see Pisa and Lucca or join a day tour that will take you by bus to Siena and San Gimignano. It's also possible to get to Siena and SG by public trans, but moving 5 people around might be easier in a tour. That's your call. You know your family best.

When is this trip? Considering you'll need multiple rooms, I wouldn't leave hotel booking too late.