My personal first was Elfen Lied, but although I loved it (and it still ranks as one of my personal favourites), it may be a little heavy for a first-timer, mainly because of the over-exposure of gore, and the rampant fanservice. Both of which I liked personally, but still...
As others have said, Death Note would be a good choice. Otherwise, depending on your favourite genre of TV show or cartoon, just pick the first one from your favourite genre that you hear about and go for it. There is no ideal place to start watching anime. Just go from wherever you want and wherever you feel okay with it. After all, anime is nothing more than the exact same as US or UK cartoons like Family Guy, The Simpsons, or the like. The only difference is the art style and the original language, after all (yet despite this hard truth my brother still makes fun of me for watching anime even though he loved Pokemon as a kid and watches various US animated series anyway...).
As far as subbed or dubbed go, I highly recommend you watch shows dubbed where possible. The reason is basically that subtitles mean you have to pay more attention to what's at the bottom of the screen than what the characters are doing, so you miss a lot because of having to read the subtitles. If I wanted to read the dialogue I'd go out and read manga instead. Seriously, dubbing is much better. People will complain that most anime has really bad dubbing, but that's something I've hardly seen, and in the very few cases I have heard bad dubbing it's still been infinitely preferable to missing half the action because of reading the subtitles...
