I'd call it genres rather than styles. Style is something a bit more narrow, and I'd mostly use it for something like 'music style' or 'animation style' - which are all just single dimensions of the art of anime.
As for me - I like everything, swords, romance, mecha, comedy - you name it. It doesn't make a massive difference to me.
However, to me - anime comes in two main flavours: either it entertains or it challenges. Or sometimes both.
Entertaining anime - here I mostly prefer modern-day based romantic comedies, but I'm open to anything else that I see as fun. I don't take it too seriously, and just have a good time. It's something I need to get me relaxed after those tough days...
Such as - Love Hina, Azumanga Daioh, Ai Yori Aoshi, etc...
Challenging anime - open to largely anything. It can be subtle (like Haibane Renmei) or it can be explicit (Eva), it can come in virtually any genre variety, but it's always food for thought. As observations suggest though, I handle outright headfuck and shock tactics very well, which I guess explains my incessant appreciation for Eva.
Both - something like FLCL. Avant-garde at its' best.
So, I guess I just put up a mostly-worthless rant...

The Birds are using humanity in order to throw something terrifying at this green pig. And then what happens to us all later, that’s simply not important to them…