4 cities - which order to see

9 days on the ground in Italy with 10 nights ?

With the amount of background in the question I could throw a dart at a map and give you an answer or the same stock answer people get for a first trip.

Surely you want a customised answer. So please give a little more detail. What does this teenager like? You? Why do Dolomites and Milan and Florence attract you?

Surely the airport would one that connects to your airport in Chicago. Which is that?

Four cities is a lot in 9 days - I would keep it to no more than 3 (you can even have two city bases with a short day trip to a third place) so you can have 3 days each (even then it won't be truly 3 days because travel time between them must be factored in). I'd pick 2-3 places not too far apart geographically, unless you want to have a hectic trip (I did a very hectic trip in 9 days so I'm speaking from experience). As far as where to fly into/out of, I'd select that based on price so you'll need to price it out first.

A lot of people prefer to fly into Venice, or Milan, but since you are firm for Venice, start there.

You might only want 3 stops total, especially when you consider the travelling time between them. Not only the train time, but the check-out of one hotel, getting to the train, arriving at next destination, and check-in to next hotel. You don't want to spend all your time travelling.

How old is "teenager"? Male or female? What are their interests? Very important to get their input so they feel a part of the trip.

Where you fly out of depends on where you end your trip and what is convenient. You CAN fly out of Milan, Florence, or Genoa.

3 sounds like a better plan to me but:
If you really want to see all 4 you would probably come up with something like this for a plan:
Fly into Venice 3 nights, Train to Florence 2 nights, Train to CT 3 nights, leave early as possible in the AM train to Milan Center to see the Duomo, stay nearby in city (1 night) and then fly out of Milan the next day.
That would be 9 nights, if you actually have 10 add 1 to Florence.

Not sure I would work in the Dolomites even if you added a day, I assume you will not have a car and think the bus schedules could be challenging for a day tripper to work around.

For me 9 days = 8 nights

Venice - 3 nights
Florence - 2 nights
Cinque Terre - 2 nights
Milan - 1 night

Why is it that everyone places the Cinque Terre so high on their list of places that they must visit in Italy?

Sure, it is a UNESCO World Heritage site, but that is for the way horticulture/agriculture is done there, not for the towns or "hiking".

Could be for photography reasons???
The convenience of getting there cannot be one of the reasons.

Cinque Terre are high on the list because for the past couple of decades they have been featured in most travel magazines and guidebooks (including the one authored by a guy in Redmond, WA). That is the power of marketing.

The itinerary will have to be dictated by the number of nights on the ground.

Assuming 9 nights on the ground with a minimum of 3 nights in Venice and 2 at the Cinque Terre, you could do:
Arrive at Venice (VCE)
Venice 3 nights
Florence 3 nights
Cinque Terre 2 nights
Milan 1 night
Depart from Milan (MXP or LIN)

I find it difficult to fit the Dolomites in your itinerary. The Cinque Terre are out of the way wherever you come from.
The Dolomites would be good if you ditched the Cinque Terre and maybe even Florence and do only a northern Italy vacation. You could insert lake Garda in the picture, instead, if bodies of water and fishing villages are important to you.

I'm gratified to see that I'm not the only one here who thinks a trip to Italy does not have to include the Cinque Terre. I was fortunate to visit it briefly in 2008, before it became as widely touted as it is now and before it became a cruise ship stop. It was lovely and I'm glad I visited. But since then, I've found many places in Italy that are at least as charming, with much more to offer besides the colorful village views. And, the reports of overcrowding make it sound really unpleasant.

LP - when are you going? July/August? How quickly can you and the teen pack/unpack, get up and get going? Changing locations eats up a lot of time in addition to the journey times. You need to plan door-to-door, with time to get (with luggage) to the station and find your train. How are you counting the 9 days? What time do you land? Figure several hours to get from the plane to your hotel before you can start sightseeing. Figure that you'll arrive jetlagged and sleep-deprived and you may be dragging on Day 2 and even Day 3, until you adjust. What time is your departure flight? You need to allow plenty of time to get from the hotel to the airport and arrive 3 hours before takeoff.

I suspect your only non-stop flight option is Rome. If you skip Rome, you will have connecting flights in both directions. If you change planes in the U.S., you may have to change terminals in both directions (domestic/int'l). You will have to collect your luggage and go through passport control and immigration before connecting to Chicago. Be careful that you don't choose a route that requires you to change airports (like NYC could be).

We went to the CT on our first trip to Italy in 08 because of...ahem...a certain travel show host/travel book author who shall remain unnamed. But I'm sure you can guess. And for the record, we really enjoyed it. Not so much our second visit in 2012. Would I even know about CT were it not for the unnamed author - most likely not (well, I would have learned about it eventually, but it certainly wouldn't have been on our first trip itin).

With 9 days, personally, I'd stick to 3 places. You could always do Florence and day trip into Tuscany region or Pisa, or do Milan (seem to be lots of flights from there) and day trip to Como area...

I went to CT in mid November 1999 and had a true "back door" experience. My return in late October 2010 was underwhelming.
Too crowded in the towns, hiking trails with fees and gates are against my religion. For me there are much better places to spend precious Italy days. I don't get itineraries with more time in CT than Florence, Venice, etc. I'm not a CT hater it just doesn't work for me anymore. As Yogi said' "Nobody goes there anymore it's too crowded".

My suggestion would be to omit the Cinque Terre on this trip. There seems to be a perception that the "trails" connecting the towns were intended for hiking, when in fact the paths were created so that farmers could access the terraced vineyards and olive groves from the towns.

The towns themselves are not that unique, and are almost completely devoted to the tourist trade. You will struggle to find a shop that sells anything very useful, such as an electric drill, bag of cement or coil of wire.

It would seem that Mr Steves sees the Cinque Terre area as a "not to be missed" place in Italy. But when you have to stand in line to walk along a path, maybe that area can be missed, at least when one's time in Italy is brief.

I went to the Cinque Terre the first time maybe in the late 1970's. I have gone practically every year in the 1980's and a few times in the 1990's and 2000's.
It was great in the early years. But the place got more and more crowded by the day, especially starting with the late 1980's.
I visited the last time in early May 2009. It was ridiculously crowded and do not plan to go back anytime soon unless some American friend or relative begs me to take them there.
If it fits in your itinerary, visit it at least once. But don't go out of your way to get there.

Thank you all for your feedback. The teenager is male and 17 yrs, old and we have both been to Italy previously but to Rome and south. I will consider all the comments about potentially reconsidering 3 stops vs 4. Our trip will be 10 nights, 9 full days. Again appreciated all your comments!

Considering the feedback, thinking perhaps drop Cinque Terre but now thinking adding Switzerland if instead we stay in Milan. Perhaps a day trip into Switzerland is possible and worth it?????

What do you want to see in Switzerland? You will be close to Lake Como. That is easy to get in/out of.

I will be in Lake Como region for a wedding (actually Lake Garda), so I was thinking of going elsewhere. If staying in Milan, thought a day trip into a swiss border town might be nice.