The Empire is gone. From its ashes comes the First Order. Leia's Resistance, supported by the Republic, is trying to stop the First Order from finding and killing her brother, Luke Skywalker (the Last Jedi) who is missing. That's all covered in the opening crawl.
The focus is on the First Order, because figuring it out it important to the story. It relates to Finn. The Resistance is slightly less important because it relates mostly to Poe, who is MIA for most of the film. When he comes back into the picture, the Resistance gets its highlights. By then the Republic's capital and fleet have been blown up, and all the exposition on them needed for the plot has been taken care of by Hux's speech.
The Republic is a non-entity for the rest of the movie because it can't do anything in the short term. It is mentioned that because the fleet is gone, the Resistance is all that stand against the First Order at that moment. And they don't have a lot of time seeing that they know Starkiller Base is going to target them now since they know the Millennium Falcon can be tracked (they can probably track it to Luke now as well).
This makes the Republic and knowing anymore about it than we would have from the PT, unimportant. What little is presented make it seem like after the Empire fell, the Republic was restored. That was the goal of the Rebellion. So most of the information on the Republic was presented in the PT and what little we see of hear about it supports that the Republic is essentially the same Republic we are introduced to in TPM, pre-Chancellor Palpatine.
Rey's story doesn't relate to any of the political parties. Her story balances on the Force and personal relationships with people she meets. She grew up in the wrecks of the previous war (as far as we know anyway...she seemed fairly young when left there) The larger world overwhelms her, but First Order, Resistance, Republic....they still mean nothing to her or her story. Only the Force, the Light and the Dark, are starting to mean something. Everything else is her deciding what path to take. Go back to Jakku, run from her powers, or move forwards, and complete the plot of the film....find Luke Skywalker.
This is her story. Until those factions mean something to her, they won't matter much to the story.
For Luke, the Empire gained meaning and the Rebel Alliance gained meaning in ANH. The Empire through them killing him family, and Alliance through Leia.
For Anakin....well, TPM wasn't so much his story, so we'll guess Padme and say that the sides mattered a great deal to her since the invasion was of the world she ruled and the Republic was a do nothing state aside from the sometimes help from the Jedi. For Anakin, it becomes important in later films because of Padme.