Travel
You will be most likely coming in by plane, as railway connections between Spain and the rest of Europe are not many. Here is some information that may be useful to you if you are:
Flying to A Coruña
There just a few international flights to A Coruña (Paris, Dublin, Milán and Geneva), so you will probably be coming from the two major airports in Spain, Madrid or Barcelona. Don't worry, the layover isn't usually long, and when you get here, you can take a bus directly to the city.
When you arrive to the airport, you can get to the city by taxi (expect to pay around 16€), or by taking bus A4 (1,60€, to be paid in cash), which can take you to an stop where you can take the urban bus. This bus runs every half an hour from 7:15 to 22:15, with extra departures at 6AM and 11PM.
Flying to Santiago de Compostela
Santiago de Compostela has a much bigger international airport, as it is Galicia's capital. If you fly directly to Santiago (there are more international destinations available).
When you get to Santiago de Compostela, you can take line 6A from the airport directly to the train station (1€, paid in cash), where you will find frequent trains to A Coruña, for as low as 6€ and half an hour of train travel. Just be sure to buy your train ticket in advance, as otherwise you may not get one, in which case you will be able to travel by bus, conveniently located just besides the train station.