By train
Arriva Trains (http://www.arrivatrainswa...) run services to Caerphilly from Penarth via Cardiff Queen Street and Cardiff Central stations.
By bus
Bus number 26 (http://en.wikipedia.org/w...) starts from Cardiff Bus Station and goes to Caerphilly. In fact you can cover Castell Coch (http://en.wikipedia.org/w...) on the way to Caerphilly, because the bus stops very close to Castell Coch in the village of Tongwynlais (http://en.wikipedia.org/wiki/Tongwynlais), on it's way from Cardiff to Caerphilly. A 'Network Dayrider' ticket £7.00/adult will give you access to all Cardiff buses plus all buses in South East Wales, including Bus 26, for the day. A one way ticket just for Caerphilly will cost lesser.
By car
Caerphilly is located just a few miles north of Cardiff, on the A469 road.