This is a difficult one and not the first time it has been asked.
There are few easy and cheap ways to get directly from the Algarve to the Costa da Sol, Valencia and Barcelona.
I would suggest that bus would the fastest and cheapest even though it will take the best part of the day.
There are two buses per day from Lagos to Seville. (Estacion das Armas Bus Station)
See: http://bit.ly/9abIPU
Alsa.es - Seville (Estacion das Armas marked without *P*) to Valencia Timetable
http://bit.ly/9JMgx6
Hope this helps
