I would certainly hope that reading through the topics at https://www.ricksteves.com/travel-tips/transportation/trains would leave you not-stumped on how Eurail passes work.
The way to find the fastest connections between any two towns served by train is through the DB link at Looking Up Train Schedules and Routes Online. You can enter your specific Swiss starting point and Cinque Terre destination town. Some summer schedules are not yet published to/from the Cinque Terre, so look at a date before June 10 for best results.
A Eurail pass covers all trains in the areas you'll travel, except from Interlaken to Lauterbrunnen and further up that dead-end mountain valley, but it gives you 25% transport discounts instead (which do not count as travel days on a flexipass; use them throughout your stay there; we would sleep in Lauterbrunnen, Muerren or Gimmelwald, not Interlaken).
You could do this trip with a Eurail Global youth pass for 5 travel days, plus somewhat expensive seat reservations for London-Paris and Paris-Basel (or a cheap one for Paris-Strasbourg; I'd book both of these from home, since they can sell out) and a cheaper one or two reservations on the route from Interlaken to your Cinque Terre destination (easy enough to book at a train station in Europe). Currently, you can buy 6 travel days for the price of 5 at $335 per person. But you don't have plans to need 6 days and the last leg from Cinque Terre to Florence is only about a $25 value.
This price won't necessarily be cheaper than buying separate tickets. If you book the first two, more expensive trips a month or several in advance, then you'll get the best prices. Info about when you're going would help you get a better answer. Maybe we can assume summer, if you're currently in college. That should give you enough time to catch some advance-purchase discounts.
To go with those tickets, I'd consider either a Swiss Travel Pass for 3 or 4 consecutive days or a one-month Swiss Half-Fare Card. These can start working from the border at Basel and give you more coverage around Interlaken. See details on our Swiss rail page. The Italian tickets can again be bought in a Swiss station, where staff can easily calculate the rate from the border.
