I noticed when looking at the wikipedia page for Season 3 of GoT, that Shireen and her mother (who's name is escaping me ATM) are new characters. So does that mean that they skipped over the prolouge from ACoK?
Eh, kinda. Elements of it are incorporated, but no, in S2, Shireen doesn't appear and Selyse only as a background extra standing next to Stannis.
Honestly, I'd recommend just focusing on reading the books for now and worry about how the show changes or adapts certain things when you get there. You may end up getting spoiled for both looking around Wikipedia and the like.