I understand that the north is Siberia, the west is desert, and south is mountains, but was there ever a time a Chinese leader tried to expand beyond these lands? If they didn’t , why not? Was China rich enough in resources that there was no reason to?
You're rather confident that they didn't. Under the Qing especially, there was much movement of Han Chinese into regions outside China proper. See this answer for details. Moreover, there has been significant movement of Han Chinese people around Southeast Asia: to take just one example, Singapore is majority-Chinese. See this answer by /u/Itsalrightwithme for more. And these answers don't even cover the colonisation of southern China from the Qin Dynasty onward – hopefully someone can come in with more on that.