I just started GK3, and I have pretty much 0 idea whats happening. Gabriel started with a card that has 'Prince Jame's' card, I called it, and inferring from the conversation it seems somebody's son has been kidnapped? Is this supposed to be related to whatever happened on the train in the intro?

I feel so lost that I can't help but feel I'm missing some material thats supposed to set up the story and let me know whats happening? Did Sierra include some 'story-only' manual or insert that isn't included in the GoG package?

The original game includes a graphic novel which contains the backstory of the game (who is Prince James, what is Gabriel's task, how he got on the train and became unconscious etc.). You can find the graphic novel in PDF form here:

http://www.sierrahelp.com/Documents/Misc/Gabriel_Knight_3_-_Graphic_Novel.pdf