Portugul

You didn't say where in Portugal you will be starting from to get to Sevilla. The most economical way to get to Sevilla is by bus from Lisbon. Check the Alsa Bus Line website for the itinerary so you'll have an idea where to catch it. Last year, the cheapest air fare I found from Lisbon to Sevilla was through Vuehling airlines but then, their flight takes you from Lisbon to Barcelona and then back to Sevilla. They offer super discounted fares if purchased in advance. If you need a place to stay in Lisbon, check the Hotel Florescente, very convenient to transportation, and in the heart of Lisbon. Don't forget to try the Bacalao dishes and the Pasties de Belem when in Lisbon. Enjoy your trip!

Based on your post, I would highly advise against going from Portugal to Sevilla for "a day or so." Sevilla is about an 8 hour bus ride (no train connections) from Lisbon. You can fly much faster of course but at far greater expense. It sounds as if you may not have been aware of how far away they are and thought it could be a convienent day trip.

Sevilla is fine as a day trip from some parts of Portugal, but I wouldn't count Lisbon among them. If you're staying in the Algarve, the closer to the Spain border you are staying the easier it is to see Sevilla in a day trip. Tavira, near the east end of the Algarve, is a great town and a trip to Sevilla could easily be done from there. The farther west in the Algarve you stay, the longer the day trip becomes. Not sure if bus or trains connect towns in the Algarve with Sevilla; it might be necessary to rent a car.