There is such a wealth of available TTRPG games that you could choose from many options but I’ve become very fond of Mothership from Tuesday Night Games of late for the quick character generation, easy play, and suitability for either one shot scenarios (where you play a single, self-contained scenario) or sandbox campaign (open-ended, improvised, discover-as-you-go serial adventures). By default it is survival-horror (think Alien, Event Horizon, The Thing, et cetera) but can easily accommodate a cyberpunk aesthetic. The one thing that might make it more intimidating for new players is the lethality and the stress mechanic which drives the players into desperate action, but the game invites rules hacking and you can tone that down if you want more survivability. It lacks a default setting per se but there is plenty of third party material from trifold one-page adventures to sandboxes and even whole-system hacks, so if you are up for creating your game world out of bits and pieces it’s a ball.
In the old but still great category is Traveller, which is regarded as the first space opera RPG. The current version produced by Mongoose still uses the basic 2D6 dice mechanic of the original game so you don’t have to invest in a bunch of polyhedral dice and it has an inexpensive Starter Set to try out for cheap. The default setting is more classic ‘sci-fi’ in a large imperium but again it would be easy to incorporate cyberpunk elements into it as there are already rules for robotics, cybernetic enhancements, and computer hacking.
Stars Without Number provides a more open ended setting and a mix-and-match set of capabilities and rule options, and has a free version.
I’ve never played it, but Cyberpunk Red (and its predecessor, Cyberpunk 2020) are quite popular. If you want to mix in some fantasy, Shadowrun has a devoted following.
If you are really interested in a highly developed transhumanist/post-human apocalypse setting with plenty of hacking, body mods and enhancements, ‘uplift’ soecies, and even the ability to shift consciosness into a ‘morph or fork it off into a virtual copy, Eclipse Phase is outstanding but the rules can be complex and the setting so dense, especially for players not steeped in transhumanist ‘lore’ that it may be inaccessible to casual players, but the artwork, character options, and science and technology are virtually unlimited.
I’m not big on games based on film and television franchises, but if you are (and it does make it easier for some players to get into the game), there are several including well-regarded Blade Runner and Alien RPGs from Free League, both with starter sets available for a cheap tryout before investing in rulebooks and supplements. Green Ronin has an The Expanse RPG if that’s your bag.
As forhow to run a game in a science fantasy, space horror, or cyberpunk type setting, there are some differences compared to classic heroic fantasy type games but the fundamentals of story structure elements are fundamentally the same, and there are numerous books and Youtube videos on the topic. I recommend starting with Seth Skorkowsky as he provides good general advice for gamemasters (and players), and runs several of the systems mentioned above as well as Call of Cthulhu, which is often regarded as a very difficult cane to run for its often elaborate investigative scenarios and potential to kill or drive insane an entire party for making one wrong move.
Good luck, and have fun!
Stranger