Reid's book recommended above (below,depending on your perspective) is certainly your best bet for a ground floor introduction. An older resource that's certainly worth checking out is William Fulton's Algebraic Curves.
In connection with Fulton's book,a new resource has recently popped up online that we ALL need to check out and I'm hoping to turn all math majors and graduate students on to it: Last year at MIT, Micheal Artin taught a basic course in algebraic geometry from Fulton's book using only his own course in algebra as prequisites. Artin composed a detailed set of lecture notes and posted most of them in PDF. I'm really hoping one day he turns them into a book-and if it becomes a widely-used resource on the web,he just might. The course and these materials,including a PDF downloadable version of Fulton's book,can be found at http://math.mit.edu/classes/18.721/. The fact that Artin is actively seeking email feedback and corrections on the notes strongly suggests he's at least considering turning them into a book-the more feedback he gets,the greater likelihood this will occur. So please-everyone find time to do this,particularly the algebracists and algebraic geometers in here!
More difficult but still very accessible, is the 2 volume second edition of Shafaravich's Algebraic Geometry text.The text is very rigorous,yet very concrete-it has many pictures and examples and builds to the language of schemes rather then throwing the student immediately into these very deep waters. Most of the initial focus is on the "classical" geometry of curves and varieties.
At this point,most people would recommend the classic by Hartshorne,which has given 2 generations of graduate students nightmares.It IS as difficult as people say,but it's very well written and if you're serious about AG,sooner or later,you have to read it. You should be ready to try it after Shafaravich and a good course in commutative algebra (a la Atiyah/MacDonald or Eisenbud).
But before you break your head on that book,there's 2 other options at roughly the same level I'd recommend first. First is Mumford's The Red Books Os Of Varieties And Schemes.This is a very visual yet abstract treatment that I think you'll find much easier going,even though it doesn't cover as much. You'll definitely find Hartshorne a lot easier after Mumford.
The other resource I'd recommend is still very much a work in progress,but it's so beautiful and wonderfully written,i'd be remiss in not recommending it. It's Ravi Vakil's lecture notes, now in thier 2nd or 3rd version of the increasingly popular course he's teaching at Stanford. They can be found in thier most recent iteration at his blog;earlier versions can be found posted at his webpage. I think eventually these notes-after a few more years of polishing-will supercede everything else for graduate students on modern algebraic geometry. They are almost "Hartshorne Explained".They also contain so many insights,it's incredible. Take at look at them,please. You'll thank me later.
That should get you started-good luck!