I'm just going to assume you don't want to watch subbed anime, seeing as you're just now getting into anime. Here's a few good entry anime with solid english dubs divided by series.
MECHA: Mobile Suit Gundam: The 08th MS Team, Macross, Code Geass (similar to death note).
MYSTERY: Monster, Attack on Titan, Fullmetal Alchemist.
ACTION: Cowboy Bebop, Samurai Champloo, Fate/Zero.
PSYCHOLOGICAL: Serial Experiments Lain, Ergo Proxy, Texhnolyze, Neon Genesis Evangelion.
MOVIES: Kara no Kyoukai series, Colorful, Stranger, Redline.
If you're able to watch subbed anime, which means Japanese audio with subtitles then I highly recommend you watch Legend of the Galactic Heroes. It's a long anime, but it's without a doubt the best I've seen. I highly recommend using this website called MyAnimeList, it lets you sort through anime and what the community ranks them and you can find new anime that way. Feel free to add me on MyAnimeList and leave a comment if you need anymore suggestions.
My MAL -
http://myanimelist.net/profile/Rance-sama