When did Judaism first begin to show up in Ancient history? What were its major influences?

In the days of Abraham, in a place called Ur which was Chaldean and now is Iraq. The major influence was Y'WH who is creator God. Others have since been an influence but not usually for the good.

I thought it was quite early, but I couldn't remember the Pharaoh's name, but I found it for you:
http://www.gnmagazine.org/issues/gn12/israel_archaeology.htm

"...in 1896, British archaeologist Sir Flinders Petrie found evidence of Israel's existence as far back as 1200 B.C., precisely the time of the events in Judges. In the ruins of an Egyptian temple, he discovered a monument that narrated the military victories of Merneptah, an Egyptian pharaoh. In this beautifully carved pillar, dated around 1207 B.C., the monarch mentions the nation of Israel.

For this reason the monument, technically termed a stele, is called "the Israel Stele." It can be seen in the Cairo Museum. On it Merneptah recorded his victories in Canaan and mentioned Israel as one of his vanquished enemies. This would place the battle during the time of the judges of Israel, when Israel was continually being attacked and invaded by nearby peoples and then liberated by the judges God chose and used to deliver His people.

In the last two lines of the text, the stele mentions four of Merneptah's defeated foes in Canaan: "Ashkelon has been overcome. Gezer has been captured. Yanoam was made non-existent. Israel is laid waste, [and] his seed is not."

The reign of Merneptah is dated around 1212-1202 B.C. By recording his victory over Israel, Merneptah shows that during this time the Israelites were already in possession of the central portion of the land."

What were its influences?
Culturally, the Jews came out of Iraq (they were Semites) and spent 400 years as slaves in Egypt. Yet their religion bears no resemblance to either country's. This is attributed to the fact that Abraham received a direct revelation from God to leave His homeland in Ur and go south to Canaan, there to be the founder of a people who worshipped one God, the creator of the whole earth.

Hope this helps.