Eurail timetables

Hi Vincenza. Yes, the schedules change on that date. However, they will be 95% the same, so until the new schedule is posted, you can look at schedules for the same day of the week a month or two out from now to get the idea. p.s. your title mentions 'Eurail'; not sure what web site you are using, but trenitalia is the company that runs most of the trains; suggest that web site.

As Bob implied, Eurail doesn't run trains, but sells rail passes. In your case, you'll need some tickets on the high-speed Frecce trains. Buy them as soon as the schedules for your travel dates become available. (Don't forget to use the Italian originals on the Trenitalia website!) Advance reservation comes with considerable discount.
I hope you realise that Venice as a day trip means at least four hours on the train. And Assisi involves at least some five, if not six hours.

I think most railroad companies will come out with the new schedule about a month and a half before the change, but sometimes Trenitalia hasn't shown their's until after the change. Still since the new period covers summer and fall, it's not much of a change from winter and spring; just a few adjustments.

Eurail doesn't run trains, but they do show schedules (but no fares) on their website, but they also have links to the national railroad websites, which have fares.

Florence to Cinque Terre

You can't get a schedule or train to Cinque Terre. Like French Riviera or New England, it is the name for a region. You need to specify one of the towns at which the train stops.

You can't get a schedule or train to Cinque Terre. Like French Riviera
or New England, it is the name for a region. You need to specify one
of the towns at which the train stops.

Those towns would be Riomaggiore, Manarola, Corniglia, Venazza or Monterosso. It will very likely involve a change of train in La Spezia. Do not use the Eurail site for trains in Italy; use Trenitalia or Captain Train, or Italo (for trains between large cities only):

http://www.trenitalia.com/tcom-en

https://www.captaintrain.com/en

http://www.italotreno.it/en

Thank you to all whom have responded. Yes, pardon my use of the word Eurail. I was using Trentitalia schedules and was panicking about not being able to get beyond June 12. Thank you for easing my panic attack!

For fast trains, it is very likely you will see them as they become available for purchase 120 before you travel dates. Regionale trains later, since they are negotiated between Trenitalia and their local district governments. There are subsidy's involved here, but there will be plenty of trains serving the Cinque terre district as it is a popular area.