Catania airport is the most convenient having direct daily bus connections with Milazzo in the summer (check times with Giuntabus, but usually about 4pm) - about 3 hours journey.
From
Palermo Airport, take a bus or train to the city centre and then take the train from Palermo direct to Milazzo. Be warned that the last train leaves Palermo around 6pm.
Reggio Calabria airport is geographically the closest, but is not well connected for international flights and requires a ferry crossing to Messina and then a bus (or train) to Milazzo.
Also worth considering, is
Lamezia Terme. Lamezia is 2 hours train journey from Villa San Giovanni - from where you can catch the ferry to Messina and then a bus or train to Milazzo.
Milazzo has it's own station, but be warned that it is about 3km from the centre of town (we don't advise you to walk this road in the dark) and therefore requires a bus or taxi onward into town. If you are tied to public transport, you may well therefore want to consider arriving by bus.