For starters, I recall that at least some editions of the trilogy included several appendices after
Return of the King. And, as already mentioned here,
The Silmarillion contains a great deal of background on Middle Earth, but I didn't finish it either! I don't think there's anything wrong with it, but it is written in a rather dry, textbook style that can be a letdown after the dramatic pacing and personal involvement of the trilogy itself. From what little I remember, there are some juicy bits about the origins of the various races, the holders of the rings, and why someone always seems to be muttering a sad poem about the moldy ruins that inevitably crop up on windswept crags.
One of the things I like about the LOTR (and other fantasy/sci-fi epics, like
Dune and Robert Jordan's massive
Wheel of Time series) is the feeling of complete immersion that comes from a thoroughly fleshed out world and history. Many of the obscure references made in the books are not just tossed in as filler, but have actually been detailed elsewhere. You can certainly enjoy the story without them, but if you like, it can be interesting to go back and make the connections later.
For an interesting, if somewhat far-fetched analysis of LOTR from a psychological standpoint, you might try
The Individuated Hobbit: Jung, Tolkien, and the Archetypes of Middle-Earth, by Timothy R. O'Neill. I didn't agree with all of the author's theories, but it was a good read, and an easy introduction to Jungian psychology.
Finally, Tolkien drew extensively from western and northern European history, particularly the pre-Christian cultures like the Celts and Vikings, so it wouldn't hurt to brush up on the anthropology and mythology in those areas.