Well, you can always do both: a lot of anime started out as manga, so you can try which one suits you better. I, for example, love both, however, I spend more time watching anime than reading manga. I find that it interests me more, especially as there is music involved, colours, it's dynamic and you can actually listen to the characters - somehow it makes them come alive.
Then, you can also choose whether to watch dubbed (mostly in English) or subbed anime (Japanese voices with English subtitles). I prefer subbed, but many people enjoy dubs as well, so if you don't like, you can always try the other. I believe that many anime series offer both.
If you're looking for a place to watch them online, I've got a good website. It works in my country, but I don't know about you, so you might want to check it out. It's called "kissanime"
Where to start....
If you're asking me, I'd say, start with anime movies. They have an excellent story, they last for about one hour and a half and they will easily draw you in. After you start liking them, it will be a piece of cake with the rest. I recommend watching anything created by Studio Ghibli. They have some amazing anime films out there and some of the most popular titles are:
Tales from Earthsea
Howl's Moving Castle
The Cat Returns
Spirited Away
I could name many titles, but these are my personal favourites.
If you'd still rather choose an animated series, then yes, Death Note is as good as any. Some people have also mentioned Naruto. That one too is very good to begin. If you're interested in mystery, you might also like Ergo Proxy or perhaps even Psycho Pass. If you're more into magic/fantasy, Fairy Tail is among the most popular nowadays.
The best is to go to "Anime List" (if you decide to visit the "kissanime" website I'd mentioned), pick a genre and go browsing. That's how I find most of the anime series I'm watching. True, sometimes I don't like them, but sometimes I do.
