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.
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 with a cruise ferry: 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 French national ferry company SNCM (bear in mind that they strike a lot more than the other one) http://www.sncm.fr/
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.
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: thanks to EasyJet, Bastia and Ajaccio are now connected to Paris Charles de Gaulle international airport.
Bastia to Basel, Geneva, London Gatwick, Lyon, Manchester, Paris CDG, Toulouse
Ajaccio to Basel, Geneva, London Gatwick, Lyon, Paris CDG
Ryanair has just started flights from the southern airport of Figari to the low cost airports of Paris Beauvais and Charleroi (Belgium).
http://www.ryanair.com Figari to Paris Beauvais, Charleroi
Germanwings flies from Bastia and Calvi to Cologne/Bonn airport, and from there you can connect to other German cities. https://www.germanwings.com/