Some people do use "Geronimo!" as a sort of battlecry when plunging into hazardous (or just exciting) situations. The Doctor may have heard someone else use it in his past, remembered it and, in his current incarnation, taken a fancy to using it himself. If so, the guilty party is probably Ace, who used "Geronimo!" that way while she was travelling with the Doctor (see Battlefield). The fez may also date back to the same period in the Doctor's life (see Silver Nemesis).
In later uses, he's generally said it to refer back to that first time.
Behind the scenes, both Steven Moffat and Matt Smith have said that they never intended it to be a repeated 'catch-phrase', but Moffat said it seemed so appropriate to use it the second time, and Smith had so much fun saying it, that they couldn't resist doing it again.