Rogue One and A New Hope, the Galactic Civil War hasn't quite escalated to a place fans are more familiar with, but the movie's worldbuilding does a good job at establishing just what's happening in the larger universe and based on the non-movie canon, we can piece together the state of the Empire and the Rebellion during the events of Solo: A Star Wars Story.
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The Empire
There's not a whole lot of mystery about the Empire in this era. As the predominant force in the galaxy, its story has been told pretty widely in a variety of non-movie mediums. Solo picks up about 6 years after the fall of the Jedi and the reformation of the Republic into the Galactic Empire. There are still clone troopers in service, although, as we see demonstrated early on, the Imperial military has already shifted to focus more on conscription. Not only has Jango Fett's original DNA sample been diluted over the years, decreasing the quality of clone output, which was an expensive venture as it was, but the Empire also saw conscription as an efficient means of subjugation. It forces citizens of occupied planets to have some skin in the game and identify as a part of the Empire instead of having it simply be an occupying force.
Behind the scenes, things are as sinister as always. Darth Vader and the Inquisitors are tracking down Force-sensitives and other dissidents, and Wilhuff Tarkin is overseeing the Tarkin Initiative, where the Empire's secret weapons, including the Death Star, are being developed. While the Death Star had seen significant progress under the supervision of Orsen Krennic, Galen Erso had gone into hiding, taking many of the secrets of the kyber crystals with him. Without this knowledge, the Death Star's superlaser couldn't be completed. Sometime during the events of Solo, Krennic finally tracks down Erso and forces him back into the Death Star project.
Technically the Empire is still under the rule of the Imperial Senate, but Emperor Sheev Palpatine had already sufficiently undermined their power through the course of the Clone Wars. While the governing body still had some influence on galactic events, there was very little concept of checks and balances and Palpatine was mostly free to do as he wished, keeping the Senate in place mostly as a means of maintaining order.
The Rebellion
For all intents and purposes, there is no Rebellion yet during the events of Solo. At least there's no Rebel Alliance, and any organized effort to start one is mostly in small pockets. Bail Organa and Mon Mothma have been working behind the scenes to forge alliances since before the fall of the Republic, and those networks have expanded to include a few rebel cells, but outside of the sharing of intelligence and maybe the exchange of resources, the network remains silent as Mon Mothma pushes for more diplomatic solutions in the Senate.
Ahsoka Tano has ed up with Bail, taking the name "Fulcrum" and serving as a chief liaison with the growing network of rebel cells. There are pockets of resistance all over the galaxy, though, a part of Organa's network or not. Occasional small victories draw the Empire's attention, such as when an assassination attempt on the Emperor and Darth Vader results in the destruction of a Star Destroyer over Ryloth, but the rebel threat is disorganized and small enough to not draw a significant focus. The former the former Jedi Caleb Dume, now undercover and going by the name Kanan Jarrus, meets Hera Syndulla, the daughter of Cham Syndulla, who orchestrated the destruction of the Star Destroyer on Ryloth. Together, Kanan and Hera lead an uprising on a small mining world, resulting in the destruction of another Imperial flagship and the loss of Count Vidian, an important Imperial efficiency expert responsible for maximizing industrial output to major projects like the Tarkin Initiative.
Han himself comes into with one such fledgling rebel cell led (in a twist) by Enfys Nest, although, once again, Enfys is und with a larger Rebellion. Still, the coaxium they steal in the Kessel run becomes the nest egg Enfys needs to build the cell up proper.
All in all, Solo has by far the lowest investment of any Star Wars film when it comes to any sort of larger galactic struggle, instead focusing on the darker corners of the galaxy, and, specifically, Solo and his crew. Even so, the backdrop of larger galactic politics is impossible to ignore. Solo may steer away from the conflict that would eventually become the Galactic Civil War, but the threads that will eventually come together in Star Wars Rebels, Rogue One, and the original trilogy are still more than evident.
