By train
There are routes from Portugal and cities in Spain best one is probably from Madrid, although the service is slow and the choice of times is not the best. The introduction of a high-speed train due in approx. 2018 will come to drastically ease train access: There will be fast connections from and to Madrid and, later on, Lisbon and Porto. There's a narrow gauge link between Ferrol and Asturias, operated by FEVE.
By ship
Although you cannot travel directly from the UK to Galicia by ferry, you can get pretty close by taking the ferry from Plymouth to Santander Brittany Ferries or Portsmouth to Bilbao P&O Ferries .
Santander is in the region of Cantabria on Spain's northern coast. It lies between Galicia's neighbour Asturias and the Basque country. Bilbao is a bit more faraway but still worth the trip following all the coastline from the Basque country to Galicia.
By car
Road communications from and to Galicia are quite good, with plenty of motorways connecting Galicia with Spain and Portugal.