I'd highly recommend Dota 2 to a new player into the genre. For one thing, Dota 2 is basically the same as the original Dota, with updated graphics on a superior engine. For another, it's completely 100% free in every way that effects you performance in game- all champions unlocked from the start.
LoL (League of Legends) is also pretty newbie friendly, but I can't recommend it because you can't really even play the real game until you grind literally for months to level your summoner up to 30, and even more time if you want to unlock champions. This process can be accelerated by paying cash, but why do that when you can play Dota 2 for free? I did enjoy LoL quite a bit before Dota 2's beta release, but now that both are easily available I don't see much reason to play LoL anymore.
Also, even though both games are "free", Dota does require a steam account, which means it's much more of a hassle to create a new smurf account. What this means for you, as a new player, is that when you start playing Dota 2 you will actually get matched up with new Dota 2 players more often than not. LoL on the other hand makes it a little bit too easy to create new accounts, which means a lot of the low level new players are really smurfs who are just trying to go for easy pub stomp victories.