kelli,
I'm sure there are trains in June from the locations you mentioned. Could you clarify which specific towns/stations you were using as the search criteria? Note that the Cinque Terre is not a destination, but rather a group of five small towns. It's necessary to specify which town you're departing from, and also which Florence station you want to arrive at (Florence has several stations), and Firenze Santa Maria Novella is probably the most common.
That particular route will most likely involve at least one change, possibly at Pisa Centrale. Most of the trains will be Regionale so there's really no benefit to buying the tickets in advance as the price is the same either way. If you buy the tickets at local stations in the Cinque Terre, you must remember to validate Regionale tickets prior to boarding the train on the day of travel, or you'll risk hefty fines which will be collected on the spot.
If you could provide more specific information, I'm sure the group here will be able to get this sorted for you.