how to split our time between Rapallo and Cinque Terre

This is a matter of personal preference of course, but I think Rapallo is too big of a city to be considered as quaint as the Cinque Terre. If it were my trip I wouldn't spend any time in Rapallo. Moneglia maybe. It's worth a visit, but you can go to Rapallo or Santa Margherita very quickly from the Cinque Terre by train. Stay in Monterosso.

I would also keep Cinque Terre as the home base. It does sound too similar, and too nearby, to warrant picking up and moving to Rapallo (which I have not visited). If 5 nights here is too much, is there another stop that would benefit from adding a day?

I'm heading to Cinque Terre for my third time this May. I love that place! Never been to Rapallo. When are you traveling? The Lemon Festival starts on May 17th in Monterosso al Mare. Party time! We always stay at least 3 days in Cinque Terre. There's plenty of hiking and great boat trips among the five villages as well as Porto Venere. You definitely will have plenty to do and see.

Stay in the Cinque Terre. If you love the outdoors, I would recommend Riomaggiore or Manarola not Monterosso or Vernazza. Those are party towns. Riomaggiore & Manarola are more real, certainly with more to do (other than shopping) than the others. In Riomaggiore you can rent kayaks, and you can begin hiking the trails at the southern most point. For swimming, none of the CT beaches have sand. Monterosso has the best beach, but the most tourists to go with it.

"None of the CT beaches have sand"?? Per Wikipedia, on Monterosso: "The beach is the only extensive sand beach in the Cinque Terre."

Thank you for all of the great responses! We are booked to stay in Monterosso, mainly due to the fact that there is a beach, but also because at this point in time, good accommodations in July are hard to find in Cinque Terre :)