I wasn't raised in any religious tradition (except insofar as you could consider my parents' committed atheism a "religious" position). I don't really believe in god but I don't believe in no-god. it's just never been part of my life. But I am really isolated and missing a sense of community. I also struggle a lot with depression and I've considered "finding religion" but I don't really know where you start with that. forgive me if this is a weird question. just something on my mind right now.
edit: i'm not asking abt any specific religious tradition. I live in a big american city so i'd probably have a variety of options but it seems to me that the most likely places would be churches of any denomination or a mosque.
Short answer no. There have been many times non-Jews have attended synagogues I’ve been at. I would say the proper etiquette would be to inform a congregant, an usher if there is one, of your interest or curiosity, just so you’re included in the appropriate parts, but not offered responsibilities or honors that are reserved for members/adherents of that faith tradition.
I’m speaking in general terms, but typically in Judaism there are some prayers or Torah readings that are reserved for Jewish people, and there is a minimum quorum of 10 Jews (Men in Orthodoxy, adults in the liberal denominations) required for some practices. You would want to let someone know to prevent you being counted as the “10th.”