The region is divided into the départements (counties) of Aude, Gard, Hérault, Lozère, and Pyrénées-Orientales. You can fly to the regional capital Montpellier (Hérault) by low-cost airline, or to Carcassonne (Aude) and Perpignan (Pyrénées-Orientales). The flight time is roughly 2hrs from London.
A more environmentally friendly option for family holidays in Languedoc-Roussillon is the train: take a Eurostar to Lille and change to a TGV to Montpellier, Bézier, Narbonne or Perpignan. London–Montpellier is about 8hrs 30mins travelling time, meaning you can arrive in plenty of time for dinner if you take an early-morning Eurostar. Alternatively, take a late-afternoon Eurostar to Paris then transfer onto an overnight sleeper (to Narbonne or Perpignan). Prices start at £109 return, £94 children 4–12, £99 ages 12–25. The Man in Seat 61 (seat61.com) has all the info you could possibly need; you can book via Rail Europe.
Montpellier is a long old drive from ferry ports (1040km/646 miles from Calais; 987km/613 miles from St-Malo in Brittany), although of course there are countless lovely places to stop over en route should you have the time and inclination. One quite expensive time-saver is to put your car (and yourselves!) onto the Motorail train from Paris to Narbonne (or from Dusseldorf in Germany). You could also shave off some driving time by sailing from the UK to northern Spain – the ferry ports of Bilbao and Santander are 677km (420 miles) and 774km (480 miles) from Montpellier respectively.