Does anyone know the best (fastest) way to go to Seville from Malaga airport?
Is there a place to leave luggage at the airport or at the train station?
Thanks!

It depends on your arrival time at Malaga airport. In journey time terms, train is fastest. You take the electric train from the aiport to Malaga RENFE station (about 10 - 15 mins) and change platforms for the train to Seville. See the RENFE website for times. (No need to enter a date at this stage.) However, as you'll see, you could just miss a train and have a longish wait till the next.
The alternative is to take the bus from the airport to the city bus station, change platforms and take a Continental Autos/Alsina Graells bus to Seville. The bus takes longer than the train but it's a more frequent service.
See this for more info on the transfers from the airport to the bus and train stations in Malaga.