Cinque Terre to Antibes?

For train schedules, use German Rail http://reiseauskunft.bahn.de/bin/query.exe/en or Trenitalia http://www.trenitalia.com/ (click on the British flag in the upper right for English). A quick look for the randomly chosen date of June 26 shows the fastest train from Monterosso to Antibes as 5:40 with two changes. If you rent a car in Italy and return it in France, you will have a very high drop charge (other posts indicate €500-600), so unless you are returning to Italy with the car, you probably don't want that. Cars returned in the same country they are rented from usually have no extra surcharge, even if it's a different city.

The Trenitalia site won't show Ventimiglia-Antibes because the trains on that leg are operated by SNCF (French National Rail). The German Rail site will show complete timetables for the entire route. Taking the bus won't work very well. First you would have to get to Genoa by train where you would catch a Eurolines bus to Nice. Then you would have to take either the train or a Lignes d'Azur regional bus to Antibes.

Thank you all so much! This is all very valuable information! I really appreciate you taking the time to share your knowledge. Cheers,
Tony