Answer: Atop a high mountain
The Jade Palace was the ultimate Dojo of Kung Fu, hosted by the greatest Kung Fu teacher, Master Shifu. Tigress, Monkey, Viper, Crane and Mantis were Shifu's students training with him on the dojo (although they were nothing like a newbie. They were the Furious Five, the five fiercest Kung Fu warriors in China). Po became the trainee inside Jade Palace as a result of Master Oogway's decision.
There was obviously no Jade Cave in the Valley of Peace, and of course, the Palace was nowhere near the main protagonist's house. It's really, really far, in fact. There was a lake, however, named The Lake of Sacred Tears. But it was pretty small, and the Jade Palace was not inside it.