Valencia International Airport is served by low cost airlines from most of European cities. Most intercontinental flights arrive at Barcelona or Madrid in Spain and many other major European cities.
Travel from the airport to Valencia city center takes ~ 20 min by taxi (costs ~ 25€).
There is also an Aero-bus from the airport to city center (tickets: €2.50) and a direct underground line connection to city center. The metro station is at the ground floor of the regional flights terminal (tickets: €3.90) (see map at http://www.metrovalencia.com). The journey takes around 25 minutes.
Detailed information on travel to and from airport can be found at http://www.valencia-cityguide.com/tourist-information/transport/transport-from-the-airport-to-the-city.html.
Alicante International Airport is also served by many flights from most European cities including low cost airlines such as bmibaby or Easyjet. A taxi from Alicante Airport to the Alicante train station (costs ~20 €) where a Euromed train can be taken from Alicante to Valencia (cost. ~40 € return, ~ 2 hours trip)
Barcelona International Airport El Prat is also served by many flights from most European cities including low cost airlines. A direct train can be taken from El Prat Airport to Barcelona Sans Train station. From there the Euromed train can be taken from Barcelona to Valencia (cost. ~70 € return, 3 hours trip)
There are trains every hour from Madrid (Ave from Atocha) or Barcelona (Euromed from Sans Estació) Train Station to Valencia Joaquín Sorolla. The journey time is 1.40 h from Madrid and 3 h from Barcelona (it is highly recommended to book and buy your train tickets in advance, electronic tickets available at http://www.renfe.com
There are also trains every hour from Alicante train station to Valencia Estación del Norte (journey time ~ 2 h).