If you're reading this, you probably know this, but I just whacked together a list of popular games and interesting quickstarts so I may as well put it here!
A lot of games have advice on how RPGs work, but a good (and free) non-game specific example comes from game designer and author Greg Stolze, How to Play and How To Run Roleplaying Games, at the bottom of his download page.
Check out free demo or "quickstart" sets for genres you're interested in. These often contain sample adventures as well as cut-down versions of the rules (They usually have no character generation system, for example, providing ready made pregenerated characters instead.) They should let you try out gaming in general as well as the specific systems.
Fantasy is dominated by Dungeons & Dragons - quickstart and learn-to-play adventures for the current edition here - and its variant/rival Pathfinder. D&D has the advantage of name recognition and a lot of sourcebooks and adventures, including many downloadable free from here - the ones not marked the the red and white "subscriber only content" icon. Pathfinder has no quickstart, but lets you look at the whole basic game online here.
There are many other options, though. I started with the much simpler gamebook-based Fighting Fantasy - there's an Advanced version which is still a lot simpler than D&D in any form - quickstart here. There's The One Ring, based on Middle-Earth and designed to create adventures that feel Tolkienesque. There's Warhammer, from the same world as the wargame - demos for the current version here - and Warhammer 40,000 games as well.
The classic horror game is Call Of Cthulhu, based on the H.P. Lovecraft mythos - and its demo adventure The Haunting is a classic too, worth checking out even if you don't like the system or setting.
Modern horror, and supernatural PCs, feature in The World Of Darkness settings. There's the Classic WOD which recently celebrated its 20th anniversary, and the "new" WOD which is almost nine years old and undergoing a major rules update. A Nightmare At Hill Manor is the most recent nWOD mortals quickstart, a one-session "haunted house" adventure, while Reap The Whirlwind is the preview demo for the new Vampire: The Requiem core book Blood And Smoke: The Strix Chronicle. The cWOD quickstarts don't come with adventures (and use a heavily modified version of the actual rules).
For zombies, the market leader is All Flesh Must Be Eaten - demo PDF here - which covers different genres with added zombies in its various supplements. A typical one-shot zombie movie style of game, where the players' characters aren't likely to see another adventure next week. will also work with other horror RPGs like Dead of Night (no demo, but it's a small self-contained book).
The biggest current post-apocalypse RPG is Gamma World, which runs on a variant of the current D&D.
Steampunk has quite a variety as well, as demonstrated in this thread from earlier in the year. I don't really know them myself.
SF has about as wide a variety as fantasy, from licenses like Star Wars, Star Trek, Firefly and Doctor Who to original settings running for decades across multiple editions. The biggest is Traveller. Shadowrun - quickstart for the current edition here - mixes cyberpunk and fantasy to make a very gamer-friendly hybrid.
There are also games for superheroes, pulp adventurers, urban fantasy, historical fiction, comedy, romance, crime capers gone horribly wrong... Hopefully you can find something that catches your eye. Good luck!