wendy,
The railways do a schedule change twice yearly, and the next one will be on about June 12th. It would seem they haven't uploaded the new schedules yet, so that's the reason you're only seeing results to June 11. The only thing to do is keep checking and eventually the schedules will be posted.
If any of the trains you'll be using are Regionale, there's absolutely no reason to buy those in advance. There are no discounts for advance purchase and it's just as easy to buy them locally (preferably the day prior to travel). However, DON'T forget to validate Regionale tickets prior to boarding the train on the day of travel, or you'll risk hefty fines.
Are you starting in La Spezia, or coming from one of the Cinque Terre towns?