I am a dyed-in-the-wool Harry & Hermione fan! Some multi-relationship stories are fun but, for the most part, a simple monogamous H/HR is fine for me. No apologies, I don't care much for Molly, Ron, Ginny, Draco, or Severus and would be fine with any story that simply ignores their existence. Dumbledore I can take either good or bad; both are fun and can lead to good stories.
The stories I read tend to be of the romance/action/mystery genres and rarely will I start an incomplete story. I really don't have much time for angst, horror, or other deep/dark stories. There is enough of that in real life and I like to read for escape and hope to finish reading in a happier mood than when I began.
I have read many AU, super Harry stories and they are fun. I have also read several where Harry is normal and magic is only a minor topic; these too are fun. I'm not much on cannon especially if it has to do with book five (OOTP) and beyond. I think that JKR may have been a bit pressured to turn out the last three years and kind of lost her way.
My favorite stories tend to be pre-Hogwarts through forth year. I like the innocence of first meetings and learning magics. I also like the founders, heirs, and Merlin fantasies.