Michael,
I'd also strongly suggest using train rather than rental car. As the others have suggested, take the shuttle to Milano Centrale and then buy your ticket to Vernazza.
Note that your train may also access the Cinque Terre via La Spezia rather than Genoa. you'll likley have to transfer to the smaller local train.
If you know the approximate time you'll be arriving at Milano Centrale, you can research the available trips to Vernazza either using the bahn.de or trenitalia websites. That will provide you with a clear idea on the exact travel times (keeping in mind that "exact" is a relative term in Italy), how many changes and which cities you'll be changing in.
Happy travels!