I was also confused as fuck as to what the relationship was between the Rebellion, the Resistance, the New Republic, the Empire, and the First Order.
The crawl didn't really explain it, and neither did the movie either in action or exposition. I only know what it is now after reading some supplemental material. I don't think it's good for a movie to have to force your viewers into reading supplemental material in order to have them understand the relationship between organizations in the movie.
When I first heard "New Republic", I assumed that the galaxy reverted to a pre-empire state, and the old republic was restored/liberated. I was utterly confused as to why the New Republic and the Resistance were two separate things. It's like that comment on RLM's Star Wars TFA review: "Shouldn't the resistance have been like, the army???"