It sounds like the Republic was secretly supporting the Resistence. Not sure why that would have been necessary.
It would be a war by proxy.
So maybe there was some treaty where the systems that were occupied by the First Order were not technically a part of the Republic (along with other neutral self governed systems) and they agreed to let those systems be ruled outside of the Republic government. But there was a faction within the Republic that recognized that the First Order was gaining power and influence and doing something more. So the Resistence was formed with secret supplies and financing by the Republic to put them down?Kind of like that.
This is pretty much exactly it. They apparently flesh this out in the "Visual Dictionary" that was released along with the movie. It'd have been nice to get a bit more explanation in the movie, but at the same time I like movies that don't explain every little thing all the time through ham-handed dialogue because I feel like it generally hurts the narrativeto do so.
The New Republic arose from the Rebel Alliance after the Rebels defeated the remnants of the Empire at the Battle of Jakku about a year after the Battle of Yavin in 6 (that's why there's all of those giant ship skeletons on Jakku). The few remaining Empire loyalists fled to the Unknown Regions (an existing part of Star Wars lore from other stuff) and created the First Order as a successor state to the Empire.
The New Republic vowed not to repeat the errors of the Old Republic by having a large standing army that can be used to subvert control and signed a treaty with the First Order to limit fleet construction and disarm. Leia disagreed with this and argued that they should be arming up and destroying the First Order rather than letting them rebuild in the fringes of the galaxy, but the new Senate voted against this and labelled Leia a warmonger.
The Resistance arose through the efforts of Leia to build something to fight the First Order and is supported -- "off the books" -- by a significant part of the senate who provide money and materials through back channels. Thus the lines by Hucks and the First Order about the senate "lying" to the people, etc.
The senate also apparently rotates every few years to be hosted in a different system in an effort to be different from the unresponsive Old Republic, thus why it is the "Hosnian system" that is destroyed in the movie, not Coruscant.
I also think that's why Poe has the seemingly odd reaction he does when he's first captured and shoved out of the transport ship at the start of the movie. He has a little "bug eyed" moment when he looked around and sees where he is. The First Order is, by treaty, not allowed to construct Star Destroyers and as far as anyone knows, they don't have any. It is possible that the one we see in the movie is the only one currently in existence (we only ever see one on screen as far as I remember).