Could you clarify a few points.....
If I'm reading the information correctly, it looks like the overall duration of your trip is about 16 days, split between 9 days in Italy and 7 days in Ireland. Is that about right?
I also live in western Canada and have made that trip so have a good idea what your experience will be like. As this is your first time travelling to Europe, you'll lose the first day in travel and will arrive in Rome the day after you depart home.
It's unfortunate that you won't have time for even a few days in Rome, but with such a short Itinerary, it's probably better to focus on the two areas you most want to see. Allow for at least half a day (sometimes more) in travel each time you change locations.
Most of the trains between FCO and Florence require one change at Roma Termini. Your travel day is going to be very tiring so hopefully you can get some rest on the flight. There are some potentially expensive caveats to be aware of when travelling by train and other public transit in Italy. You'll encounter this for the first time on the Regionale train from FCO to Roma Termini. If you need further information, post another note.
Determining how to portion your time between Florence and the Cinque Terre will depend on what you want to see in each location. If you don't have one already, I would highly recommend packing along a copy of the Rick Steves Italy guidebook to help in your planning. Which of the five Cinque Terre towns are you planning to stay in?
As you're planning to head for Ireland after your Italy visit, the Cinque Terre is a good choice for your last stop, as it's reasonably close to the airports in both Pisa and Milan.