A great fast way to find out all the destinations from a particular airport is the airport's Wikipedia page (a trick I learned on this Forum). Here's the page for Barcelona: https://en.wikipedia.org/wiki/Barcelona%E2%80%93El_Prat_Airport#Airlines_and_destinations
A quick look shows that Vueling or Ryanair fly to Barcelona from Seville, Granada, Jerez, Alicante, and Almeria.
To find flights on the day you want to go (not all flights are daily), use Skyscanner: https://www.skyscanner.com/
Be sure to do a dummy booking on the airline's own website to learn all fees. Vueling and Ryanair have lots of extra fees for all kinds of things, and enforce their rules strictly to maximize revenue - don't be caught by surprise.
High speed trains (AVE, for Alta Velocidad Española) in the south start at Seville or Malaga, then meet in Cordoba on the way to Madrid and Barcelona. AVANT trains are almost as fast, on shorter routes (such as Seville to Cordoba). Of course, you can always take a slower train or a bus to connect to the AVE; the bus network in Spain is more extensive than the train network, and the buses are much nicer than Greyhound in the US, so they're a perfectly good alternative.
Here's the map of the current (in green) and future (other colors) AVE routes: https://en.wikipedia.org/wiki/AVE#/media/File:HighSpeedSpain.svg
Note that from Seville to Barcelona, the train is probably as fast, if not faster, than the plane end-to-end, since you don't have to get to the train station 90 minutes early and the train stations are much more central than the airports.
If you are taking the AVE, book in advance for best fares (can be 1/3 of the last minute price if you book far enough in advance). But booking Spanish trains can be tricky. The RENFE site does work, as long as you use PayPal rather than a credit card. You should start with this tutorial from TripAdvisor: https://www.tripadvisor.com/Travel-g187514-c80518/Madrid:Spain:Buying.Renfe.Tickets.Online.html