You can probably just watch "Silence in the Library"/"Forest of the Dead" and then skip ahead to Eleven's first episode without missing much in the way of continuity. But yeah, I forgot to mention that if you're going to watch Eleven's episodes, you definitely need to have seen that pair of episodes first, since River Song is so important to seasons five and six. Without those, the show won't make much sense.
That said, you should definitely watch at least most of Nine and Ten's episodes (I agree that "Love and Monsters" is skippable; that episode was putrid. "Fear Her"'s not that bad though).
For what it's worth, I started out with Eccleston's first season and never regretted it.