The best way to get around Seville’s compact city centre is definitely on foot. If you want to visit anything out with the city centre, then the city bus network will get you to your desired destination. The main bus stops are at the Plaza Ponce de Leon, the Prado de San Sebastián or La Barqueta.
A single journey is affordable at around 1€ for a single fare. Leaving you some change for tapas when you get there.