Walking is the best option in Seville. All attractions are close by and you do not need public transport to reach them. In any case there are lots of narrow roads and alleys, so buses are limited on where they can go.
Only Triana is slightly away from the historic centre but still easy to access on foot or by bus. If you need transport, just use the bus and tram. If you are in the center I recommend the tram as the ride is very nice. In the tobacconists and kiosks you can buy a rechargeable card to use in the buses. I highly recommend it as it saves you good money.
Bus and metro are the best travel options within Seville. Within the city center, you should use the tram. It is modern and convenient for major Seville attractions.
Tram coverage in Seville
The Seville tram covers major attractions located in the city center. Seville tram stops are San Bernardo, Prado de San Sebastián, Puerta de Jerez (close to Torre del Oro with nice parked horse carriages around), Archivo de Indias (a beautiful 18th century building free to visit), and Plaza Nueva with many shops and bars.
The tram does not go to Triana. I recommend a walk to Triana across the beautiful Puente de Isabel II bridge.