The native language of Timur and Babur was Chagatai Turkic.

Babur wrote his autobiography in Chagatai Turkic, but was also fluent (and literate) in Persian; e.g., he would write and recite poetry in Persian.

According to Ahmed Ibn Arabshah (who wrote one of the two main early histories of Timur), Timur "was illiterate, reading, writing and understanding nothing in the Arabic tongue, but of Persian, Turkish and the Mogul language he understood enough but no more." However, Ahmed Ibn Arabshah also says that he would listen to Persian histories being read, and knew them well enough to correct errors made by the reader, suggesting his Persian was quite good, rather than just functional.

As for Genghis Khan, he spoke Mongolian as his native language. I don't know of any good sources on other languages he spoke, but Rashid al-Din notes in an anecdote about Ogedei that Ogedei claimed that Genghis Khan spoke only Mongolian (and not Turkic). As for literacy, it is usually believed that he was illiterate, but it's possible that he was literate in Mongolian, and also possible that he was literate in Chinese. For some of the debate on this, see David C. Wright, "Was Chinggis Khan literate?", Studia Orientalia 87, Helsinki 1999, pp. 305-312.