Do I play it's actions on my next turn or my oppnts. turn? Seeing as the Kingdom and quest phases take place before my Capital phase.

Card reads : Kingdom. Action: when this card enters play, gain 1 resource.

Quest. Action: When this card enters play , draw a card.

or does it not work untill I play a second DerricksBurg Forge?

You get the resource/card on the turn you play it.