What is the best way to get from Paris, France to Cinque Terra, Italy?
Fly to Pisa or Genoa (whichever is cheaper or offers better times, both are similar in distance to the CT)
From there take a train to town in Cinque Terre you are staying in
This is my first time to Europe. Is there a particular airline you would suggest?
Ryanair and Easy Jet are the bargain airlines that serve this route. Be sure and be aware of luggage policies.
Look at Skyscanner for flights.
This is my first time to Europe. Is there a particular airline you would suggest?
no. just stick with direct flights and follow the rules with the baggage
These are the only direct flight choices:
Easyjet to Pisa (PSA)
or
AirFrance (operated by HOP!) to Genoa (GOA).
I wouldn't bother with Ryanair to Pisa, because to catch Ryanair you'd need to go all the way to Beauvais airport, which is quite a distance from the city of Paris.
When is this trip taking place?
Your best bet on that route is EasyJet from Paris / Orly to Pisa. For transport from central Paris to Orly, have a look at this website....
http://www.parisaeroport.fr/en/passengers/access/paris-orly/public-transport
Orly is a smaller airport than CDG, and I find it much easier to manage.
When you arrive at PSA, you can take the automated People Mover into Pisa Centrale, and buy your tickets there for your trip to the Cinque Terre. You'll need to specify which of the five towns you're going to when you buy the ticket.
It's likely that at least one of your trains will be a Regionale, which means you MUST validate your ticket prior to boarding the train, or you'll risk hefty fines which will be collected on the spot! You can check all the rail solutions on the Trenitalia website.....