As JK mentioned, The Last Jedi tells us very little about Snoke. But things we do learn are that:
Snoke commands the First Order from his massive command ship. It is instantly recognizable to members of the Resistance.
If he rules from any particular planet, there is no hint as to what it is.
Furthermore, Snoke is
an extremely powerful Force user. He effortlessly suspends Rey in the air and drags her from place to place in his chamber. Most impressively, he drags General Hux across a floor from perhaps many parsecs away, during holographic communication.
reads the thoughts and emotions of Rey and Kylo with ease and bridges their minds across a vast gulf of space.
We also learn that
Snoke first influenced Ben Solo to join the Dark Side during the latter's training under Luke's tutelage. This may also have occurred across a great distance.
In addition, we discover that
Snoke employs a squad of eight Praetorian guards (clad completely in red). Previously, Palpatine also had Praetorian guards in his service.
From dialogue between Snoke and Kylo Ren, we learn that
Snoke is quite preoccupied with nurturing the "next Vader" and is deeply disappointed by Ren's inability to rise to this challenge.
Finally, perhaps the most interesting thing we learn:
"As detailed in the [film's] Visual Dictionary, Snoke wears a ring with an obsidian setting from the catacombs beneath Darth Vader's castle on Mustafar. But the truly interesting tidbit is that the gold ring is etched with glyphs of the Dwartii, presumably a reference to the Four Sages who served as controversial lawgivers and philosophers from the dawn of the Galactic Republic."
This last point may suggest something about Snoke's origins or motivations — although nothing is for certain.