cynthia,
The easiest solution and the one I'd probably use is to take one of the Chamonix Shuttles to the Geneva airport first thing in the morning on the day the tour ends (Friday?). It's likely there will be others from your tour needing a Shuttle, so you may have company on the trip. Your Tour Director will help you arrange that.
easyJet has flights that depart at 12:05, arriving Pisa at 13:15. From Pisa Centrale it's an easy trip by train to the Cinque Terre. The airport is very close to Pisa, so the trip is a short one. Depending on how quickly you can get from the airport to Pisa Centrale, there are two departures you could use....
- Depart Pisa at 13:42, arrive Monterosso at 14:52 (time 1H:10M, direct)
- Depart Pisa at 14:48, arrive Monterosso at 16:16 (time 1H:28M, 1 change at La Spezia)
I used Monterosso as the destination to research these trips, as you didn't specify which of the five towns you'd be staying in.
If booking with easyJet, be sure to read their Terms & Conditions carefully, especially related to luggage rules. If you want to confirm the trip to the airport, you might contact the RS tour department.
Unfortunately, websites such as Rome2Rio aren't always able to "think outside the box" and cover all eventualities, so they're limited by their programming. I travelled from Chamonix to the Cinque Terre last year, so did quite a bit of research on the transportation options. If you'd prefer to travel by train and spend a couple of nights at an interim location, I can offer a suggestion.
Bon Voyage!