How are non-Jewish people expected to behave when they enter a Jewish place of worship?

Answer 1

Jews are always happy to welcome a non-jew into a synagogue, but there are one or two things that could be done that really endear you to your hosts: In a synagogue, always wear head covering. Anything will do, but the best thing is the traditional "yamulke". Often synagogues have a supply of them at the doors. If you are still worried, best bring a Baseball cap or something similar. If you are going to an orthadox synagogue, try not to bring any electrical items, especially mobile phones, as their use is forbidden on the Sabbath (though as an outsider you would be readily forgiven) If you are going in on a Saturday, the traditional greeting is "Shabbat Shalom" - literally Sabbath Greetings. If someone says it to you, just repeat it back to them.

Answer 2

Respectfully. The same goes for any house of worship, regardless of the religion. Different congregations have different styles. Some are more into decorum, some are not. The atmosphere in some small traditional congregations can seem quite disorderly, with people praying at their own speed out loud, coming together only at key points in the service. Larger congregations tend to be more decorous, as are many liberal congregations.