Theresa,
For travel to different countries, use the Bahn.de website. Trenialia usually only shows trains which originate in Italy. The Bahn.de site covers most of Europe. Also, as someone else mentioned you'll need to use the name of the Cinque Terre town as your starting point, as the "Cinque Terre" is not a rail stop.
That's going to be a LONG trip to do in one day. Using Monterosso as the starting point, it looks like trips range from 10-12 hours, with 4-6 changes. One of the easiest trips is a departure from Monterosso at 07:05, arriving Mittenwald at 17:35 (time 10H:30M, 5 changes at La Spezia Centrale, Firenze Campo di Marte, Bolzano, Brennero and Innsbruck).
Is there any possibility of staying at an intermediate location for a night or two, to break the trip up?