As an introductory adventure, IIRC it advises the PCs not start as a member of any faction. This way they can be introduced in play, allowing the players to decide their opinion of them for themselves.
Factions are a wonky, and because of that a lot of DMs either downplay or ignore them. You can technically be a member of any number of factions, but since each has their own goals and motivations, maintaining allegiance to just two would be really hard, and three all but impossible. In general, their primary use is in DDAL (D&D Adventurer's League) which is organized play.
You can't automatically tell if someone is a member of a faction, but for cooperative play, it's advised that this information be public. Many wear/carry/have a token or icon that indicates their faction, such as the Harper's pin, but the average person probably wouldn't know the significance of it, even if they noticed it in the first place.
I use the same general rules for factions as a form of Reputation. If you do something for a noble, you'll gain some Reputation with that noble. The same for a church, city, guild, etc. This allows me to quickly reference the response NPCs and groups might have with them, depending on their relationship with anyone they have Reputation with. Oh, and I allow for negative Reputation, putting them and their allies as Hostile towards the character.
