louise,
First of all, welcome to the Forum! As I'm sure you've noticed already, there's a very well travelled group here that will be able to help you plan your first trip to Italy. I have a few thoughts to add.
My first suggestion would be to have a look at the RS Italy guidebook, as there's a wealth of information there that will help your trip go smoothly. If you want to buy a copy, the 2020 edition will be available in Dec. 2019 but it will probably only have minor changes from the 2019 version.
Given the time-of-year you'll be visiting Italy, I'd probably stay with the original plan to start in Rome and fly out of Venice as the weather will probably be warmer. My suggestion would be to keep the Cinque Terre on the list as it's an easy route to travel Rome > CT > Florence / Tuscany > Venice. April should still be reasonably "civilized" in the CT so you shouldn't have to contend as much with the cruise ship hordes. If you're out touring the area during the day, you should be able to avoid most of the crowds. Despite the negative publicity, it's still a beautiful and somewhat unique area, and a UNESCO World Heritage site. You'll have to decide which of the five towns you'd like to stay in (my favourite of the five is Monterosso). If you do decide to stay there for a few days, I would highly recommend getting a hotel booked as soon as possible.
Whether you'll need a car or not will depend on which places in Tuscany you plan to visit. The larger towns are well served by rail or bus service so if you only plan to visit larger towns, public transit would be a better option. As mentioned in the link above, note that each driver listed on the rental form will require the compulsory International Driver's Permit which is used in conjunction with your home D.L. IDP's are easily available for a small fee at any CAA office. You'll also need to be extremely careful to avoid the ZTL (limited traffic) areas which exist in most Italian towns. Each pass through these will result in hefty fines, which you likely won't know about until several months after you return home. Do not drive in Florence as the city is just about saturated with automated ZTL cameras!
There are also some potentially expensive caveats to be aware of when using trains and other public transit in Italy. You'll need to do some research on that. Details should be in the guidebook but I haven't checked recently.
With a 17-day time frame, you should easily have time to cover all the locations on your original list. You could (for example), structure it somewhat like this.....
- Fly inbound Rome - 5 nights (including day trip to Orvieto or other nearby location)
- Travel by train to Cinque Terre town of choice - 3 nights
- Travel by train to Florence - 5 nights (including day trips to Lucca or other places)
- Travel by train to Venice - 4 nights (possible day to Padova)
There are lots of possibilities so this is just one suggestion. I can elaborate on this if you need more information.
In addition to travel and sightseeing plans, it would be a good idea to do some research on the aspect of pickpockets and scams in Italy, as it's likely you will encounter that.
Good luck with your planning!