We will be on a cruise that is stopping in La Spezia and we plan on taking the train into Cinque Terre.

Any advice on what villages not to miss and the best way to visit this area?
We will be in port from 7:30am until 7:30pm (on shore and all aboard times)
I know we will not be able to visit it all but a good overview would be great

TIA

How much physical exertion do you like? It is not the villages so much as the walk between them that is the appeal.
You get off the train, buy a park pass, and walk between the towns of your choosing. Some walks are more strenuous than others. You could walk from one end to the other, then train back to the one you like the most for lunch.
What month will this occur?

This will be in late June and cannot be too strenuous. We have one person in our party that can walk well on small inclines but a lot of steps are a problem

Then there are several stretches you will need to avoid, but the Via dell'Amore is not one of them--it is paved and flat.
You might look into doing a ferry trip for the views since you won't be walking--if you aren't sick of being on a boat that is!
ferry info
http://www.navigazionegolfodeipoeti....e.asp?lang=eng
trail info
http://www.parks.it/parco.nazionale....terre/Eiti.php

Do the ferries run on Sunday in the summer?