Hoo boy, this is a fun one. Personally (although I've never played it) I would say start off with 4th edition D&D. Before I get strung up and burned for my heresy, let me explain. 4e, as others have said, are a lot like video game RPGs, so that takes away some of the learning curve. Yes, it's like D&D's version of WoW, but at least it has the basics there. Explore dungeons, kill hideous monsters, loot their stuff. In-depth Story it's not, but it will give you the basics of the tabletop RPG in a way that's a little easier to understand than some other games. Once you get tired of doing the same damn thing over and over for a million play sessions, go on to other games and expand your horizons.
Personally, while I've only been playing tabletops for a few years, I've been in on D&D (both AD&D and 3/3.5), Alternity (futuristic space scifi with aliens), GURPS (oh gods the equations!), Mage and Vampire (the OLD ones, thank you; Malkavian for the gerbil!), Imagine (the most complicated, yet closest to realistic system I've ever seen), Dark Heresy/Rogue Trader/Deathwatch, Mechwarrior (A Time of War is the newest edition), and Shadowrun (which is basically a playable version of the book Neuromancer, aka cyberpunk). I've also run a couple games of Dark Heresy (and gotten the questionable compliment of being a twisted GM. Yay?)
Personally, while I've only been playing tabletops for a few years, I've been in on D&D (both AD&D and 3/3.5), Alternity (futuristic space scifi with aliens), GURPS (oh gods the equations!), Mage and Vampire (the OLD ones, thank you; Malkavian for the gerbil!), Imagine (the most complicated, yet closest to realistic system I've ever seen), Dark Heresy/Rogue Trader/Deathwatch, Mechwarrior (A Time of War is the newest edition), and Shadowrun (which is basically a playable version of the book Neuromancer, aka cyberpunk). I've also run a couple games of Dark Heresy (and gotten the questionable compliment of being a twisted GM. Yay?)