I think I'd recommend just staying put in one location as well: the villages are not far apart enough to mak much difference.
Besides the train, you can also travel by boat, which is great!
Yes, and no: the ferries run seasonally so are really only available in during high season. You can view schedules here:
http://www.navigazionegolfodeipoeti.it/home.asp?lang=eng
Also, there is no ferry service to Corniglia as it doesn't have a big enough harbor. That one must be accessed via train and small shuttle bus (or a long walk up the hill from the train station). The bus is coordinated with arrivals of the trains, and schedules will be posted near the station.
We stayed in Monterosso because of easy (flat) access to our waterfront hotel from the train station (did plenty of climbing during our three days but chose not to do that with the luggage!). All five villages are well worth the time to explore at length, though. If you're going during high season, get out of bed to see Corniglia early in the morning. Because that one is so tiny, it's very easily - and uncomfortably - overrun with groups during the busier mid-day hours. Two of the prettiest churches in the C.T. are there so be sure to stop into those.