I think Death Note is a great one to start, especially if you've never given anime a shot before.
It's one I find new fans very much enjoy as it's easy to get into/follow - it's a cat and mouse type thing, which you've more than likely seen before in other films and TV shows, so you'll be familiar with the tone and type of plot.
But like RadFem says, explore shows within genres you like for films/TV shows and you can't go too far wrong.