OP is attempting too much IMHO. It can be done, but it isn't recommended. Countless threads available on this kind of question.
Monterosso to Vernazza had 4000 people a day on the trail around Easter,for example.
Golly - we were there on 29 September last year (not in fact April) and it was quite empty (although I concede we headed out about 07:30 am). So clearly spreading out the punters over more months would seem to be desirable.

The days in each place will depend on you and your interests.If you have around 11 nights as above,I'd stick with the Big 3...Rome,Florence and Venice.If you can take more time then you can add some more destinations....
This is a good plan, IMHO. The "Holy Trinity" (Rome, Florence, Venice) is a sensible starting point for first-timers, and I'd personally give the Cinque Terre a wide berth during the summer season. All three deserve more time than you've given them, and you'll want to maximize your sightseeing time/budget versus spend too many chunks on transport, especially if you want to minimize the expenses.
I would almost certainly fly from Paris to Rome, and from Venice back to Paris ... I can't see fares being significantly more than train travel
Agree: keep the amount of luggage down and fly instead of going by rail.
What kind of transportation would be better (car, train, flight, mix)? I am hoping to minimize the expenses... Is EURAIL interesting in this case and how to use it?
Train, definitely. They will get you from from city to city the most efficiently. Passes are almost never a benefit in Italy: point-to-point tickets are the best method. Also, do not use the Eurail website to book them: use Trenitalia (website for national rail) or Italo (private line serving larger Italian cities).Or just purchase them when you get there. If you can commit to a specific date/time, you can often land good advance rates on non-refundable, non-changeable tickets but they're probably gone for the fastest trains next month. You could look, though.
Travel 2nd class: no need for more expensive ticket/carriage tiers.
What are the must see or do in Italy?
Entirely up to you as my must-sees might be your avoid-at-all-costs. Get a guidebook and spend some time with it. Make a list of what you see that interests you. I will note that to avoid wasting time standing in very long (and very hot) queues, buying advance tickets to some of them is a MUST. A few of these are the Vatican Museums and Colosseum in Rome, and the Uffizi and Accademia in Florence. Pre-reservations are required for the excellent Galleria Borghese in Rome but I'm guessing open slots are long gone for next month.
Waiting until the last minute to plan this trip has put you at a disadvantage for everything from snagging those lower-cost train tickets, to landing rooms at better, budget-priced accommodations, to doing prior research about what/how to see the things you want to see so you need to scramble here!

I chose italy because it seems to me like one of best European countries to discover, in terms of food, culture, and it is close to the mediterranean sea. I am into beaches a lot. I just thought that I could try it since I have to be in Paris in July, and I wanted to combine it with something else. But it looks like it is better to wait and organize it better for next July may be. If I can have about 2 weeks (meaning about 15 nights), would you be able to help me with itineraries that could include a mix of those types of sightseeing?
Thanks a lot for the help!
Sure,there are lots of beaches.
The ones that you can get to by public transport will be very crowded though.The smaller,quieter ones you would need to hire a car.
Best areas for beaches are Sardinia, Sicily, Puglia and the south in general.

If I can have about 2 weeks (meaning about 15 nights), would you be able to help me with itineraries that could include a mix of those types of sightseeing?
Shanez, if you decide to move your trip to next year, you have PLENTY of time to spend with some guidebooks...which is exactly what you should do. I am not into the beaches in Italy at all as I can see beaches here in the States but I can't see Roman ruins or Renaissance art in situ so my must-dos are very different. You're also not going to be doing beaches in Rome, Florence, or Milan. They aren't really great in the Cinque Terre, and they're not why you'd go to Pompeii. You may want to choose some different locations entirely if beaches are your focus?

You are right. I know that italy is not a ''beach destination'', but I just wanted to make the best of such a trip. I hear that there some great beaches in the south.
There are plenty of beaches. Most Italians spend the whole summer on one ;-)
Most foreign tourists don't though....partly because the beach is crowded and expensive in summer,partly because many tourists are more interested in the "cultural" aspects of the country.

I hear that there some great beaches in the south.
And I hear there are some great beaches in the states.
Much closer and cheaper.
And besides, your itinerary doesn't include the south.
If a beach, any beach, will do for a day or two, to decompress and process things you can't see closer to home, that's a different story.
But it will still cut into the time to see things you can't see closer to home.
There's a lot of that in Italy.
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