How did the people of Judah practice their religion?

Judaism centers around Torah-observance. This has never changed. The Torah contains many hundreds of commands, hundreds of which are applicable in exile also. When offering sacrifices was possible, they were offered; and when the Temple was destroyed, those particular commands of the Torah were suspended. It's like a man who lost his arms: he doesn't put tefillin on, but his Judaism hasn't changed.
How did they maintain their faith?
1) Reading the Torah in the synagogue and studying it at other times.2) Keeping the laws and beliefs of the Torah. Those who didn't do this, such as the Hellenizers and the Sadducees, went lost.

3) Maintaining vibrant Jewish communities, with communal prayer and study, mutual help, maintaining ties between the various communities, etc.

4) Remembering God's covenant and promise that the Jews and Judaism will never cease.

See also:

More about the diaspora

Jewish history timeline

Was Judaism altered after the destruction of the Temple?