I love the railway line along that part of the Italian coast. As it comes down from Genoa, it constantly goes in and out of tunnels, with snatched views of beautiful bays in between them.
The trains are frequent, generally reliable and really cheap. (Sit upstairs, on the double-deck carriages, for the best views). Tickets are undated, so you have to remember to validate them for travel (by inserting them into the machine by the platform entrance) prior to boarding your train.
You'll have no problems with exploring Cinque Terre by train. See Wikitravel for loads of useful information about the area:
http://wikitravel.org/en/Cinque_Terre
Other places which you might like to visit by rail include Rapallo (which is a pleasant town, with some nice restaurants overlooking the beautiful bay) and Santa Margherita (made famous by Liz Taylor and Richard Burton having lived there). The billionaire's resort of Portofino isn't accessible by train (or, indeed, by road - because cars are banned) but there are regular boats from Rapallo and Santa Margherita. The old city of Genoa is also fascinating.
Chris