The closest airport to Lucca is the Pisa International Airport, which is 30 minute train or car ride away from Lucca. If you arrive to Pisa, you'll drive 20 kilometers northeast along the SS12.
The Florence airport (Amerigo Vespucci Airport) is the second closest to Lucca and also the second busiest Tuscan Airport. If you take the train it will take you about an hour and a half and if you drive you'll arrive to Lucca in less than one hour. When driving you should take the A11 Highway 72 km west.
For travelers from Europe
Pisa is the closest international airport, and it's only 20 km from Lucca.
Airlines that service Pisa, for example are: Aero Italia, Air Albania, Air Dolomiti, Alba Wings, British Airways, EasyJet, Euro Wings, Flydubai, Ryanair, Silver Air, Transavia, Wizz Air.
Florence is the second closest airport, 72 km from Lucca.
Airlines that service Florence, for example are: Aegean Airlines, Air Dolomiti, Air France, Air Serbia, Austrian Airlines, Binter Canarias, British Airways, Iberia, ITA Airways, KLM, Silver Air, Swiss International Air Lines, TAP Air Portugal, Volotea, Vueling, Widerøe.
For travelers from outside Europe
Rome airport is the recommended arrival airport for travelers from outside Europe. It's the busiest airport in Italy and the 10th busiest airport in Europe, so you will find direct flights from countries in North America, South America, África and Asia. A direct flight provides the least risk of missing luggage, missing connections, stress over connecting flights, and less chance of damage to your baggage.
From Rome airport you'll have to take the train+bus to get to Lucca, it will take you about 4-5 hours.
If you are open to doing a connecting flight and prefer to fly, then Pisa would be the best option.