VI, VII, or X.
X's story can be a bit melodramatic, but it's still really good, and if good graphics will make it easier for you to take the story seriously, this is definitely the place to start.
VII is arguably the best game in the series, with an incredible story and lots of sidequests to take on. A truly amazing game that you will love, as long as you're not turned off by the dated graphics.
VI was on the Super Nintendo, so it's rather old-school. But its story rivals that of VII, and it will blow you away how well they can tell a genuinely emotional and dramatic story with such old technology.
Remember that Final Fantasy is more about the story than anything else; the series tells some of the greatest fantasy tales ever told. Play whichever one will allow you to enjoy the story the most, and let the gameplay grow on you.
XIII wouldn't be a bad choice, by the way. The battle system is decently complex, but it is introduced so slowly you'll have no problem getting it. The problem is that the linearity of the game and the quality of the metanarrative (which was largely sacrificed for the sake of having some amazing inter-relational character drama that absolutely makes that story) are not reflective of what most Final Fantasy games are. XIII is a great game, but I would not recommend it as as first.