Overall Malaga has a good public transportation hub, and there are new projects on the go such as the new Malaga Metro. Additions are being made to the high speed train network. You can already get to Madrid from Malaga by train in around 2 hours!
The city centre also has a good network of urban buses that connect with the whole of Malaga. For further information about urban buses and their Malaga timetables click here: https://www.alsa.com/en/web/bus/our-destinations/airports/malaga-costa-del-sol
The west of Malaga already has a train that runs across the coast to such towns as Fuengirola and Estepons.
How to get from the airport to the centre
From the airport you can get to Malaga centre either by train, bus or a taxi. The train or bus will cost only around one or two euros. Buses run up to two or three times per hour from the airport and a taxi to the centre will cost approximately twenty euros. Bear in mind that the train stops in two different locations in Malaga centre. The first is the main train station and onthe second is Avenida Andalucia, which is a main road. The bus from the airport stops at several locations in Malaga but ends at Avenida del Parque.
Getting from airport to anywhere to the east of Malaga, you have two main options, bus or transfer. Buses will indeed save you money, however if you are travelling with children or have more than one suitcase, this may not be best option for you. The bus will take you to Malaga bus station where you will get off and wait around for the 'direct' bus. A Malaga airport transfer is of course the direct and fastest way to reach your destination. With some of the better options you don't even have to pay up front see https://www.malagaairporttaxi.net
For further information see the official Malaga tourism website. http://www.malagaturismo.com