To get to Corniglia the best way is by train. Here stop just the "regionale train"; so is better to arrive in La Spezia centrale's train station (wich is South) or Levanto (which is North) with an Intercity train or Eurostar and then change train and come to Corniglia.
The from Corniglia's train station to come to the centre is possible to walk (about 10/15 minutes) or take the shuttle bus.
Is also possible to drive to Corniglia: the best road is from La Spezia and then follow the directions to Cinque terre/Corniglia.