Tickets for the AVE are very reasonably priced as well (prices range from around 50€ to about 80€ per person depending on the departure time), but admittedly it is considerably more expensive than taking the bus.
From the airport, you can take a yellow bus known as the Express Line Atocha Renfe-Aeropuerto shuttle to Atocha railway station in Madrid, where trains will depart almost every hour of the day for Seville Santa Justa station. Once you arrive in Seville, the train station’s central location makes it relatively easy to get around to most tourist attractions in the city. Another thing to keep in mind is that this is the most popular way of getting back and forth between the two cities, trains can fill up rather quickly, so you might want to book your tickets in advance just to make sure.
Santa Justa train station, like Atocha, is in the center of the city, so you will have no trouble getting where you need to go.
As a side note, there are other slower trains (ALVIA) that go from Madrid to Seville that are slightly less expensive but there are a lot fewer departure times available, and being slower trains, it will take you a bit more to get there.