Ryan Air - The undisputed king of discount flying. From Valencia, they fly to a lot of locations covering most of Western Europe: Portugal, Ireland, England, Denmark, Sweden, The Netherlands, Germany, Belgium, France, Switzerland, and 4 cities in Italy. More routes to Valencia being opened all the time.

Vueling ES - This company is new to me, and is based out of Barcelona. Their fares are low and from Valencia, they fly to Amsterdam, Brussels, Paris, Milan & Sevilla.

airberlin - Obviously, they fly to Berlin, as well as a lot of other destinations all over the German-speaking countries. They also serve some spots in Spain, such as Palma, as well as Copenhagen, Milan, Amsterdam and Lisbon.

Wizz Air Hungary - The only discount airline I could find that frequently flies to Eastern Europe: Bucharest, Transylvania, Sofia and also Milan.

Air Europa - They fly to a few vacation spots within Spain, as well as Paris.

EasyJet - I’m really disappointed with EasyJet’s presence in Valencia — they only serve London & Bristol.

Lagunair - Two spots in Spain: León and Valladolid.

Transavia - Just Amsterdam.

TUIfly - Only flies directly to Germany (Memmingen, Stuttgart, Köln/Bonn and Hannover). Maybe worth looking into booking a connecting flight from Köln/Bonn since from there they fly to most destinations in Europe.