One of my favorite shows. Love the dragons and Dani. I love the "name/title" Khaleesi. Anyways, it's getting so good. I've seen season 1 ans 2 and I can't wait for season 3 to come out. I really hope Sansa can get out of there! And I hope John is alright.
Seconding everyone who suggests you start from at least the second book! The first is very faithful but they changed some significant things from the second book, for things like time or to make for example Dany's plotline more exciting...I don't think most of their changes are bad, but you definitely miss a lot if you only watch the show.
And the one change I absolutely hate is much of what they've done with Catelyn Stark, and her relationship with Robb/their roles in the war. They basically take all her wisest decisions and give them to him, then portray her as a hysterical mother...I can't get more into it without spoiling for season 2, but if you're not that invested in Cat or Robb I don't think you're missing much if you stick to the show only! Cat's just my favorite character and it's strange that they get characters like Arya, Sansa, Tyrion, and Cersei so completely amazingly right, but don't seem to understand or care about Cat as a character at all.
(Partly I think it's that, in the books, Robb does not have POV chapters and Cat does, whereas it feels like in the show, they wanted to age him up and make him less of a young, out of his element boy king and more of a Heroic Leader, when his whole character in the books is a deconstruction of that? So probably some of that involves making his plotline seem more centered in his POV than in Cat's, so I do understand why she gets shafted to some extent, but it's just my big complaint with the show, when everything else is so flawless!)
tl;dr you'll be missing Lyanna and Cat stuff if you don't read the books, but overall you can watch season 1 and basically get most of what you'd get from reading the first book. Where I think some of the themes in Cat and Dany's storylines have been lost with the changes the show has made, AND book 2 is where GRRM seemed to really hit his stride with pacing and whatnot in his storytelling (book 2 and 3 are the best, I'd say), so it's a good place to start. And after that you'll probably be so hooked you'll just keep reading till you're through book 5, waiting desperately with the rest of us for the next one to come out xd