How did Judaism develop from the 300s B.C.E. through the 100s C.E.?

Answer 1

Judaism centers around Torah-observance. This has never changed. The Torah contains many hundreds of commands, hundreds of which are applicable today too, even after the Destruction in 68 CE.
To answer the question:
While the Torah, its commands and the principles and beliefs of Judaism remain unchanged, Judaism has evolved as circumstances have necessitated.
1) After the end of prophecy (340 BCE), the canon of the Tanakh was sealed by a special Sanhedrin (Rabbinical court). It was the same Sanhedrin which placed our prayers in their permanent form (Talmud, Berakhot 33a).
2) Purim (355 BCE) and Hanukkah (165 BCE) were instituted after the relevant events.
3) Certain fasts were instituted in connection with the Destruction of the Temple.
4) After the Destruction (68 CE), the laws of sacrifices were suspended.
5) When circumstances made it impossible, the New Moon was no longer proclaimed by testimony; rather, the fixed Hebrew calendar was instituted.
6) The Talmud was put in writing when it became too hard to be learned by heart.
7) Seven formal Rabbinical commands were decreed. These are:
Saying the blessings over food (and on various occasions)
Washing one's hands before eating bread
Lighting the Hanukkah-menorah
The Eruv
Saying the Hallel prayer on certain occasions
Lighting the Sabbath candles
Reading the Megillat Esther on Purim.
In addition, there are many Rabbinical decrees, mostly from the Men of the Great Assembly (340 BCE), as well as later enactments.
The purpose of every one of these is to provide a "fence around the Torah," meaning to shore up something that can benefit from strengthening. An example: not handling electric appliances on the Sabbath, even if they are not connected to any electric socket.

See also:

Jewish history timeline

Was Judaism altered after the destruction of the Temple?