Welcome,
I'm an American too with absolutely no connection to the game but have been studying/watching it over the last 5-10 years or so. If you got any question, I'll try to answer the best I can and hopefully can relate to some U.S. sports - football (not soccer) in particular.
Just a couple of things that I've noticed that seem generally different from U.S. sports and football in particular that you might notice or find weird:
1) The refs: The refs seem to play a more significant part in the game and are actively telling players what to do mid-game. The refs seem generally more well-known because how they decide to call matches and what their tolerances are for certain things (esp. at the breakdown) can have a very material effect on outcomes. Also, the enforcement of certain rules seems to be a bit more arbitrary than you find in the NFL IMO. There's an argument to be made that you could call holding, for instance, on almost every NFL snap but often it's very subjective and often immaterial to the outcome of the play. At the breakdown (i.e. after a player is tackled and a ruck is formed) there seems to be a consistent violation of the laws governing the breakdown, the ruck, and poaching the ball. From my understanding, there's a balance being had between enforcing the rules and maintaining the flow of the game, so the refs have to contend with this as well. Also, if you look at scrum feeds, they seem to be crooked quite a bit. Got to admit, seems to be harder than hell being a Rugby Ref.
2) The laws/rules while generally steady, seem to be more in flux compared to the NFL. The recent NFL rule changes with the kickoff, extra point we Big precisely because changes of that magnitude are not common. In my opinion, there seems to be more rule changes in rugby that aren't merely clarifications and tweaks. From my understanding Rugby didn't really become professionalize until the 1990's so with that in mind I think a lot of the change has to do with a balancing act between creating an entertaining product, spirit of the game, and long-standing rules. Player safety as well seems to be a big one.
I'll admit this last year or so I haven't been able to watch as much so maybe some of this has changed through points of emphasis and other laws, but I just wanted to provide a viewpoint coming from a background similar to yours. It's a neat game but it definitely is a bit of a "culture shock" sometimes when looking at it through the lens of a U.S. sports viewer.