Travelling to France & Italy..10 Days...need a trail of cities to follow

This itinerary is far beyond what I would call a pleasant journey.
Each move costs you at least half a day with checking out and in of a hotel, getting to and from an airport or train station, and the actual transport time. Even if those 10 days are without your travelling days from and to Mumbai, it leaves you with precious little time to enjoy these cities.
Limit yourself to three places. I'd say Paris, Venice and Rome would make a nice program. Fly to Paris, have three full days. Fly to Venice, have two full days. Take the train to Rome for the remaining days and flying back home.
Don't wait too long with making a decision. June is a popular holiday month.

Of all the cities you list, I would drop Nice.

That is not a slam on Nice, but consider the others!

Unfortunately, the remaining four cities are still too many to visit in 10 days.

Three would be tops, and I could make the case for 2.

With just 10 days, I wouldn't do more than 3 cities. Assuming your first day is arrival day...

Venice - 2.5 days/3 nights
Florence - 2.5 days/3 nights
Rome -3.5 days/ 4 nights

Or.....
Paris - 4.5 days/5 nights
Rome- 4.5 days/5 nights. Your first day may be even shorter depending on time to travel from Paris.

Or....
Paris - 2.5 days/3 nights
Venice - 2.5 days/3 nights
Rome - 3.5 days/4 nights
You could switch this around and do 3.5 days/4 nights in Paris and 2.5 days/3 nights in Rome, depending on which city you think you'll want to do the most sightseeing in.

Keep in mind that you'll lose 1/2 day or more every time you change locations, and it's better to count the number of nights, versus days, you'll have after you arrive.

What Kathy said. This is absolutely as much as you can do in only 10 days. And I would probably limit Venice to 2 nights and add one to Florence. It takes 2 nights to get a full day in a place. spending less than 4 nights in a major city like Paris, Rome or Florence is a mistake; you end up spending all of your expensive and limited time on transport and logistics of hotel check in and out and such -- each big moves wastes at least half a day. Being there beats getting there as a way to spend time. If I had ten days for a first trip I would pick two destinations -- maybe Paris and Rome and fly between or else do just Italy and focus on Florence and Rome with maybe two nights in Venice to just walk around and enjoy the uniqueness of the city.

If you plan to do Italy, fly into Rome then immediately head for Florence by train, then head for Venice and then back to Rome and you are there to enjoy Rome and fly home.

If you plan to include Paris then fly into Paris, use a budget airline to get to Rome and fly home from Rome. Open jaw tickets cost no more than round trips.

What Kathy said. This is absolutely as much as you can do in only 10 days. And I would probably limit Venice to 2 nights and add one to Florence. It takes 2 nights to get a full day in a place. spending less than 4 nights in a major city like Paris, Rome or Florence is a mistake; you end up spending all of your expensive and limited time on transport and logistics of hotel check in and out and such -- each big moves wastes at least half a day. Being there beats getting there as a way to spend time. If I had ten days for a first trip I would pick two destinations -- maybe Paris and Rome and fly between or else do just Italy and focus on Florence and Rome with maybe two nights in Venice to just walk around and enjoy the uniqueness of the city.

If you plan to do Italy, fly into Rome then immediately head for Florence by train, then head for Venice and then back to Rome and you are there to enjoy Rome and fly home.

If you plan to include Paris then fly into Paris, use a budget airline to get to Rome and fly home from Rome. Open jaw tickets cost no more than round trips.

What Janet said. Twice.

If you plan to do Italy, fly into Rome then immediately head for
Florence by train, then head for Venice and then back to Rome and you
are there to enjoy Rome and fly home.

Janet, you don't think they could fly into Venice and home from Rome?

Thanks a lot everyone for your valuable responses.... I realized that I am trying to squeeze too many things into a short tour... ill try to increase my duration of the tour or limit a visit to maximum 3 cities. Thank you all again.