Corsica is an island, so there are only two ways to go: ferry or airplane.
The island is only 80 kms from the Italian coastline and 140 from the French coast, so going by ferry is a good option since you can bring your own car. It is also a really nice way to go to Corsica for the first time. Falling asleep after leaving a French or Italian harbor and waking up at dawn to the spectacle of the island’s mountains rising above the sea is a wonderful experience. When you arrive by ferry, you have time to get a good view of the island, to approach it slowly, distinguishing more and more details of the landscape before you set foot in the harbor.
The main harbors on the island are Bastia, Ajaccio, Calvi, l’Ile Rousse, and Propriano.
From France: ferries depart from Marseille, Toulon and Nice.
From Italy: ferries depart from Livorno, Savona, and Piombino. There are also connections with Sardinia.
The duration of the trip depends on the departure and destination harbor, and of the type of ferry (cruise ferry or high speed craft).
Cruise ferries from Marseille are generally scheduled overnight: you sleep on board and wake up in Corsica. The shortest trip is from Livorno to Bastia, only 4 hours.
For schedules and prices, visit the Corsica ferries company website: http://www.corsica-ferries.co.uk/
Or the Corsica Linea company website: https://www.corsicalinea.com/eng
From Sardinia: There are ferries connecting Sardinia and Corsica. The shortest trip is from Santa Teresa to Bonifacio, which are facing each other across the straits of Bonifacio: one hour. http://www.mobylines.com/routes/sardinia/ferry-bonifacio-santa-teresa.html
There are also ferries going from Porto Torres in Sardinia to Propriano and Ajaccio in Corsica. http://www.lameridionale.com/page/our-agencies/our-shipping-agents-66.html
From Elba: the Italian island of Elba is facing Bastia and is accessible in 1h30 http://www.corsica-ferries.co.uk/crossing/elba-ferry/index.html From the Elba harbor of Portoferraio, you can reach Tuscany in just 30 minutes.
Corsica has four international airports: Ajaccio, Bastia, Calvi and Figari. The regional airline, Air Corsica, connects them daily with major French cities as well as with Rome, occasionally.
The main airline is Air Corsica, which ensures regular connections with the French mainland. However, their position of monopoly for some destinations can make their flights quite expensive.
From/to Ajaccio, Bastia, Calvi, Figari
From/to Nice, Marseille, Paris Orly, Toulouse, Nantes, Lyon, Clermont Ferrand, Toulon, Dole, Liege (Belgium), Rome (Italy)
In the last few years, some low cost airlines have managed to gain entry into the market:
Bastia to Basel, Geneva, London Gatwick, Lyon, Manchester, Paris CDG, Toulouse
Ajaccio to Basel, Geneva, London Gatwick, Lyon, Paris CDG
Figari to Paris Beauvais, Charleroi
Ajaccio to Brussels, Geneva
Bastia to Cologne/Bonn, Salzburg, Vienna
Calvi to Cologne/Bonn, Geneva, Salzburg, Vienna