Best/closest airport to fly into and best transportation to get to Cinque Terra region?
There are many options for this. In my opinion, the choice boils down to how you can connect to either of them. Where will you be coming in from? Genoa, Pisa and Florence are all fairly close. You can easily take a train from these towns to the Cinque Terre. Check Trenitalia.com for trains schedules.
You may have more flight options if you arrive in Milan. But, you would have a longer train ride. It's about 50 minutes to Milano Centrale, then you can get a train to CT, about 3 hours.
Are you planning to visit any other cities? In my experience, it is cheapest to fly into Milan. It's a little longer train ride but worth the savings in my opinion. If that doesn't matter to you, Pisa or Florence would be closest.
We recently concluded our RS Tour with a few R&R days in Cinque Terre, and found Pisa to have the best 'nearby' options for our return to California. A puddle-jumper to Munich, then nonstop stateside.
As the others have mentioned, Pisa is about the closest airport to the Cinque Terre but flights are more limited. Genova is also close, but again not as many flights. Milan / MXP has the best flight options but is a bit further away. You didn't say where you're flying from, so it's difficult to offer more specific suggestions.
From either Pisa, Genova or Milan, you'll still have to travel by train to the Cinque Terre. You'll first need to decide which of the five towns you'll be staying in, as you'll need to know that when buying train tickets. If you decide to stay in Monterosso, that's actually a fairly easy trip from Milan as there are a new direct trains (no changes) every day.