Basing myself in Bangor, itself a pleasant enough town, I was able to visit each of the royal castles:

Conwy - by train or bus from Bangor

Harlech - by bus from Bangor or train from Shrewsbury,then a steep climb up to the castle

Caernarfon - by bus or train from Harlech to Porthmadog then by bus to Caernarfon

Beaumaris - by bus from Bangor

Harlech castle was spectacular for its location and great views.