This coming from a person that doesn't read many comics (Only recently have I started following a few, mainly DC's New 52 Batman related comics). So my knowledge is from reading info on the internet, seeing movies and watching this or that cartoon.
In terms of core, of idea, I have 2 great comics in mind. Batman and X-Men. I just love the world of batman, the detective vibe, the lunatic villains, the dark aspect, and everything that stems from it. Then I love X-Men, the class of people different from the rest, and how they cope with their powers and try to insert in society. Magneto is one of my all times favorite.
In terms of their companies, I do prefer Justice League over The Avengers, and to be honest, while I do enjoy Spider Man (And the movie Iron Man), Batman takes the lead. On a core, besides X-Men, the stories I like the most are DC's characters.