Your main options are bus and taxi.
There are many taxis waiting outside the airport waiting to take travelers wherever they need to go, but taking the bus might be your best option. There are buses that will take you from the airport to Marbella. They will be labeled “portillo” and you can buy them in advance. You can look for times and fares here. The site is in Spanish but no fret - select Amalaga Aeropuerto in Origin and Marbella in the destination. While it is easy to buy tickets online, there is no need for advance booking.
Malaga to Marbella by taxi
Malaga International Airport is 45 Km. from Marbella. The taxi will cost around €65
Most of the important towns in Spain are connected by train but not Marbella. The nearest railhead to Marbella is the town of Fuengirola, midway between Malaga and Marbella. It makes no sense to travel to Fuengirola by train and then travel to Marbella by bus or taxi. It would be better to take the bus from Malaga to Marbella