What is the best way to get from Cinque Terre to Lyon without a car? Trains or planes?
I'd take a train from La Spezia to Milan, then fly from Milan to Lyon on Easyjet. To take a train all the way from Cinque Terre to Lyon would be a long overnight journey with transfers. Easyjet will fly you non-stop from Milan to Lyon for around $50. The airports in Genoa and Pisa are closer to Cinque Terre than the airport in Milan, but neither has budget flights to Lyon.
The train journey will take 9.0hr-10.5hrs depending on the run you choose. The routing is most likely Milan/Torino/Chambery-Challes/Lyon. There is routing via Genoa/Ventimiglia/Nice Ville but it's a longer train journey.