For someone just starting and with such an open mind, you honestly have so many avenues to go off of. A fair place to start is MAL's top list
http://myanimelist.net/topanime.php
Popularity isn't always attributed to bad things. Many of the anime up there deserve recognition for one reason or another and are usually good watches for anyone starting out. After you've cleared a few you'll start to get an idea for some anime, how they work, and maybe your favourite genres that you lean towards. You'll maybe even start to branch out into some gems of their genre that are regarded in other ways, or simply flown by the general populace's heads.
I think another important thing about directing yourself towards this top list is to have at least a single other person you trust entirely to give you an idea or rundown of what most anime will entail, and why it's popular. Someone who's non-objective, who can separate their own feelings about an anime, and give you a real idea about how to progress!
It's such a neat medium and I think if you treat it with care and caress it's head gently, it'll reward you back with wondrous tales and feelings.
Alternatively Code Geass is probably a rite of passage into the anime world, whether it's considered good or not