We've put these two transport methods together as you can't get straight to Brighton either way, but there are nearby plane and ferry links which make it easy to take a trip from Brighton, or get to Brighton from the continent.
Gatwick Airport is only 30 minutes away and is the major airport south of London, making it a great amenity to have nearby. The train tickets are cheap and they run very frequently. You can also book a taxi for a set price to or from the airport, if you arrive very late at night, or have a lot of luggage.
There is a small, private use airport in Shoreham, so if you have your own plane you can fly from there!
If you want to travel by sea, you can reach Newhaven in 30 minutes by train too. Newhaven has 2-3 daily sailings to Dieppe, in France. The journey takes about 4 hours. Dieppe is closer to Paris than Calais, being a bit more south, so it can be a convenient place to start a French holiday! You can check out ferry prices over at AFerry.co.uk. They'll compare every ferry company for you so it's nice and easy!