So we are thinking of taking a train from Vernazza to Lake Como. What station/town is the one to use for Lake Como?
Varenna-Esino is probably the best touristic town on Lake Como with frequent train service from Milano Centrale.
If you look at a map, you'll see that Lake Como is shaped like an upside-down letter "Y." So there are three "arms," and distances between them are large. There are multiple stations to use for the lake, depending on which part you want to go to.
I stayed in Varenna and liked it a lot. It's "mid-lake" (the "crotch" of the Y) and from there you can take easy ferries to Bellagio, Menaggio, and some other places. Varenna is on a train line, and as Sam posted, Varenna-Esino is the station; if you want to stay elsewhere mid-lake, you take the train to Varenna, then a boat. Menaggio and Bellagio have bus service as well as boat service, but no train service.
Another place on the lake with train service is the town of Como, at the southern end of southeastern arm. I haven't' been there, so I can't comment on it. Note that Como has two train stations: Como S. Giovanni is served by Trenitalia, and Como Lago is served by LeNORD, both departing from Milano Centrale station.
From Como town, there are slow ferries and faster hydrofoils going up to mid-lake and beyond. Rick's Italy book has details on where to find ferry and hydrofoil schedules online (I don't have a recent edition, so I can't give these).