Where did judaism start and spread?

Answer 1

There are two questions here:

Where did Judaism start?
The general Jewish religious answer is that Judaism originated in Israel, but there are three acceptable religious answers according to Judaism (Israel, Harran, and Sinai). In addition there is also the historical answer (Slow Development in Israel and Babylon).

A. Religious: Israel
The general correct answer is the Holy Land (Canaan; today's Israel), since it was there that Abraham lived most of his life. It was there that Abraham made a covenant with God (Genesis ch.15) and raised his family and taught disciples to carry on his beliefs and traditions (see Genesis 18:19).

B. Religious: Harran (North Syria/South Turkey) -- The Road Between Babylonia and Israel
Judaism, theoretically, could also have developed in Harran, since it was there that the young Abraham (after being born in Ur) lived for several decades before coming to Canaan, and taught others about the One God.

C. Religious: Sinai (Northeast Egypt)
The Jewish people officially became a nation at Mt. Sinai when G-d revealed Himself to 2 million people and gave them the Torah (Exodus ch.19). This differs from other religions in the fact that the revelation involved the whole nation and not just one individual. The Islamic religious position is that this was when the religion of Judaism started.

D. Historical: Israel and Babylonia (Central Iraq)
Jewish teachings began to crystallize in the times of the Kings of Israel, but those beliefs and traditions did not crystallize until the Jewish Exile in Babylon. It was at this point, that the Torah was completed as explained by the JEPD hypothesis.

Where did Judaism spread?
Since Judaism is a non-proselytizing faith, it spread to whatever countries Jews were exiled to or were able to move to in order to escape persecution and did not spread by local conversions. At one time or another that would cover most civilized parts of the world, but primarily covered Europe, the Middle East, North Africa, and (after they were "discovered") the Americas and Australia. Judaism did not make it to Central Asia, India, China, or Southeast Asia except as small trading communities. Other than Ethiopia, Jews also had practically no presence in Sub-Saharan Africa until the colonization of South Africa by the Boers.

Answer 2

Birthplace: Judaism was founded by Abraham in the Fertile Crescent.
Abraham (18th century BCE) was born in Ur (Mesopotamia; now Iraq), where he first repudiated idolatry. He then sojourned in Harran (Syria) for several years, and then lived most of his life in Canaan (Israel).
It was in Canaan that Abraham lived most of his life, made a covenant with God (Genesis ch.15), and raised a family to be the center of carrying on his traditions (Genesis 18:19).

Abraham's family carried on his teachings voluntarily. Judaism as a binding, permanent entity, was set forth between God and the Israelite nation descended from Abraham, Isaac and Jacob at Mount Sinai in the time of Moses (Exodus ch.19, ch.24, and 34:27).All of the above places are in the ancient Fertile Crescent.

See also:Where did the Jews come from?
How Judaism was founded

Archaeology and the Hebrew Bible

Where did Judaism spread?

1) Around 2500 years ago, the Babylonians destroyed the First Temple and forcibly exiled the remainder of Israel's population to Babylonia.
While the Jews were permitted to return to Israel (Judea) seventy years later, and tens of thousands did so (and rebuilt the Temple), most of them remained in Babylonia, while others began to settle in North Africa, southern Europe, the Crimea, throughout the Near East and elsewhere.

2) In 68 CE, the Romans destroyed the Second Temple. The Romans did not force the Jews out of Judea in a single expulsion. Rather, the Romans expelled them from Jerusalem only; and the rest of Judea lost its Jews slowly, over a period of centuries, as living there became too harsh. Even then, we have records of Jewish communities who lived in Judea (Palestine) during the entire period of the last two millenia. (See:History of the Jews in Israel)

Those Jews who left Judea went to southern Europe, North Africa, Arabia, the Near East, and (slowly) further afield (especially throughout Europe).

See also:

Jewish history timeline

Where do Jews live today?

Maintaining Judaism during the exile