First, I recommend finding a group which already plays and joining them.
OP, for those who like free and feel that crunch and classic RPG feel will work for them, the entire ruleset of the Pathfinder RPG is available online for free. So what is Pathfinder? When D&D hit 4th Edition, a majority of the fanbase disliked the product. Now, I am NOT entering an edition war, but the bigger seller was the setting and ruleset for Pathfinder, which is sometimes referred to as D&D 3.75. It's a very heavy game in terms of rules, but the Beginner's Box ($30 USD) is for sale and comes with tokens, dice, and basic ruleset for characters. The rest of the rules are offered for $0 USD online from Paizo (the publisher!) themselves. Buying books is entirely optional. You can find it at d20pfsrd.com/ or paizo.com. As I said, the rules are -free-.
More free goodness can be had at Roll20.net. What's that? Why, a free virtual table top that allows you to host a campaign with all its notes. Your virtual tabletop will obviate the need for miniatures, play mats, dice, and just about anything except for character sheets. I have been playing RPGs for more than twenty years and would endorse using Roll20 whether the game is online or not. Simply put, putting Roll20 on your TV means there's less clean-up, no shortage of tokens, and every note and player handout can be controlled by the GM. It also allows dynamic lighting, fog of war, die rolls, online character sheets, and more. You can choose to pay for an account but by no means must. If you'd like to see the product, it is at Roll20.net.
Now, onto picking a game: games have two big components, fluff and crunch. A "crunchy" game has a lot of complex rules. A game with a lot of "fluff" has a lot of setting. Fluff and crunch are not mutually exclusive. Crunch by itself is like salt - you need to find how much of it you want, and that takes tasting the games and seeing how much you like.
Star Wars d6 is a super-light, super-easy to run classic Star Wars RPG. It is also called West End Games SW. Compared to most games, it's easier than easy to run. Way more approachable than almost any succeeding SW RPG. Sadly out of print, but there's eBay. Low crunch, very high fluff.
Pathfinder (their setting) is a world set in a very pulp novel feel of high fantasy. It appeals to all the classics; gothic horror, high fantasy, planetary romance, lost cities, and so on. It's also -smart-, as in the writing style is intelligent, crisp, and carefully discusses detail. It uses the Pathfinder RPG system, which is VERY high in crunch, and the setting has a ton of chewy fluff.
The World of Darkness comes in the Old WoD and New WoD flavors. Both are games about the supernatural existing in the ostensibly modern world. There are many games of these lines, so they kind of have a niche appeal to anyone who thinks Underworld or a Dr. Strange comic would make a good RPG. Pretty easy on crunch, and mountains of thoroughly readable fluff.
Call of Cthulhu is a medium-crunchy game which is about being an investigator in the Cthulhu Mythos. It's a straight up horror game where most protagonists die or go insane. This is a game for people who like the idea of being caught up in The Shadow Over Innsmouth with a mayfly's survivability. Delta Green doubles down and does Call of Cthulhu seen through the lens of conspiracy theories. It's amazing.
I cannot recommend the various Fantasy Flight Warhammer 40k adaptations. Very, very heavy crunch and they kind of depend on you already knowing the WH40k fluff, plus an unforgiving and evil game engine. You have to be a big fan for it to pay off. The games are called Dark Heresy, Rogue Trader, Deathwatch, Black Crusade, and Only War.
Unknown Armies is one of my favorite games. Super-light crunch - you can learn it in about twenty minutes. The premise is you live in the modern world and magic is real but hidden. It aims for a conspiracy/paranoia feel where anyone who learns magic is just a little crazy. It's an amazing game which oozes personality.
Shadowrun is a classic game of cyborg and magic-using mercenaries in the dystopic cyberpunk future. I can't describe it - just check out some let's plays of Shadowrun Returns the cRPG. The game has medium crunchiness and a lot of fluff. The setting has tons of personality, but that doesn't mean it's personality that appeals to everyone.
Ah, D&D4e. I will NOT take sides. The difference between D&D 4e and Pathfinder is that PF feels like a high fantasy with highly technical, rocket-tag gameplay, while D&D 4e is MUCH easier to pick up, but the game does nothing to hide it's padded-sumo MMO-esque gameplay from being at the fore. In short, PFRPG plays more like a set of rules playing out high fantasy stories and as a result is a lot more complex, while D&D 4e simplifies everything at the cost of being more "gamey" in a way that many dislike. This is a matter of taste, and while someone in this thread is going to start an edition war, ignore them. They're both good games at being EXACTLY what they are. It is your decision and the enjoyment of your group that matters. Find what you prefer.
All Flesh Must Be Eaten is a do-it-yourself, easy to run Zombie game. It is exactly what it sounds like. It's fun, cheap, easy, and doesn't ask much of you. The downside is after a few scenarios, you'll need something new. That said, it's not a bad place to start.
GURPS is mega-crunchy because it's designed to be able to do anything. This game is really for someone with the time and experience to do what they want. I cannot recommend it for beginners. The same applies to HERO system.
FATE system games such as Spirit of the Century are INSANELY easy to run and a lot of fun. They are designed for a system which has as it's goal, "Screw it, we don't need a lot of rules, let the GM make his or her calls." These are games which require a certain devil-may-care mindset where players trust the GM not to be a jerk out to get them. That requires the players to understand fundamentally that RPGs are unlike anything else they've played.
Deadlands DRIPS with personality, being a horror-fantasy set in an alternate Wild West where things went very wrong. Medium crunch and fun mechanics like dealing yourself poker hands to cast spells. It's in a good position to check out, has a nice and fun world to get into, and appeals to anyone who has a little love of horror or Spaghetti westerns in them. The product line is pretty intense, but it can be played out of one book.
I'm sure I've forgotten many and left some off the list (like Cyberpunk 2020 or Unhallowed Metropolis or Everyway or...) but that's not the point. This should give you an idea about what to look at. If you'd like to ask me anything, go ahead and do so.