There is limited bus service from Málaga airport to Sevilla.
Maria Zambrano is the main train station in Málaga. It is next to the bus station. You can take a train or bus from there. The trains and buses to Sevilla run frequently.
As Benny suggested you can take the Cercanias train from the airport to Maria Zambrano station. It is only three stops away and takes 8 minutes. The price is €1.80.
The same applies in reverse from Sevilla to Málaga.