Grab bag of info
Right, let me try pitching in a few more suggestions (mostly for Dan and Ineti) as they come to me. If I echo something someone else has already said, you can just take it as great minds thinking alike (or fools seldom differing, if you prefer ).
The Fourth Age
This is the wide open territory of Middle Earth, and you can do pretty much anything you want with it, which is both a blessing and a curse. All we really know is what happened to some of the major characters in the story, and the names of their successors, as detailed in the appendicies. There is the abortive "New Shadow" story that Tolkein started (which I think can be found in "Sauron Defeated", one of the History of Middle Earth volumes), but he never got beyond hinting at some ominous new threat to Gondor, decided it wasn't doing anything for him, and abandoned it with less than a chapter written.
Presumably you'd keep a 4th Age campaign relatively close to the end of Lord of the Rings (by which I mean within the first century of that Age). This is the beginning of a new Golden Age in the West of Middle Earth, with the restoration of the Crown of Gondor and the repopulation of Arnor. The realms of Thrandul and Celeborn expand to fill all of Mirkwood which had been under the shadow of Dul Guldor, the realms of Erebor and Dale prosper, and at some point even Moria gets restored (though the last might not actually occur durring the first century). The great evil of Sauron is unmade, the power of the Ring destroyed, the Ringwraiths undone, the Balrog no more, and Smaug, last of the great dragons has been rotting for years. There is plenty of potential for great works to be accomplished.
Still, one must keep in mind that the great theme running throughout Tolkien's work is one of fading and decay; The Golden Age may be beginning, but it's splendor does not reach that of the previous Golden Age. The magic is slowly draining out of the world; The great and terrible Enemies have been defeated, the Wizards have departed or fallen, all but a few of the great Elven lords have sailed for Valinor. Eventually the Elves, and the Dwarves, and the Hobbits, and the Orcs all disappear, and the ancient lore and dignity of Westernesse are forgotten, and we all fade, becoming Men of Twilight. The reign of Aragorn and his heirs is the "last image of the splendor of the Kings of Men, undimmed before the fading of the world," one last hurrah before myth becomes legend becomes history.
So any Fourth Age campaign would have to be more "mundane" than anything which went before. It is the Age of Men, so the great Heroes and Villains should be Men. There is plenty of Sauron's legacy left over after his fall (many of the rulers to the East and South were under his sway, and there are no doubt plenty of marauding bands of Orcs setting up for themselves in the shattered remnants of their Master's realm) that will force King Elessar to do a lot of work, but all of them are weaker, lesser successors to Sauron, just as Sauron was a weaker, lesser successor to Morgoth.
On Tolkien Scholarship
Don't be intimidated by the wealth of invented history that Lord of the Rings is based upon. If you really look as what is mentioned in the narrative itself, that history only breaks the surface in a few points, hinting at all that has come before. Unless you are planning on running a game durring the Siege of Angband in the First Age (in which case a familiarity with the Sillmarillion is going to be essential) you can get by with just the material in the appendicies and a good feel for the ambiance of Middle Earth. If your players are inexperienced with the world, introduce it to them gradually, in bits and pieces, and don't try to heap lots of ancient lore upon them. If you are inexperienced and your players are serious Tolkeinphiles, use that to your advantage; you only have to sketch out the broad details, and let them fill in the rest of the world from their knowledge. There is plenty of precident in Lord of the Rings for telling ancient lays and tales around the fire, and even Sam, (not the most versed of Loremasters by a long shot ) dredges up part of "The Fall of Gil-Galad" at one point. Really, nearly all you need of Middle Earth history leaving aside the First Age (and nearly all that has been written on the subject for that matter) is in your hands if you've got the Hobbit, the Lord of the Rings, and the rulebook.
It's only intimidating if you love it and are worried about doing justice to it. And if that's what you're feeling, then you will approach it with the care and respect necessary for you to succeed.
-Chris Landmark
"Was entstanden ist, das muss vergehen. Was vergangen, auferstehn." -Klopstock & Mahler
"Only liberals really think. Only liberals are intellectual. Only liberals understand the needs of their fellows." How much viciousness lay concealed in that word! Odrade thought. How much secret ego demanding to feel superior. - Heretics of Dune