Big Realms booster here...
I agree with the post that says to make the Realms campaign YOUR Realms campaign. But, if you're looking for at least a framework upon which to hang your own tapestry of the Realms, here's my suggestions, based on observations culled from the many Realms novels and products.
1. The Realms is High Fantasy, not Grim N Gritty. Call it Tolkein Lite if you want, just don't call it late for supper. The Realms is your classic "go to a great tavern, have a few ales, sing a few songs, get a quest, do quest, return to tavern, repeat" setting. Some think it's cliche, and avoid it. Personally, I find "grim n gritty" to be cliche these days, and a reversion to the classic style refreshing again.
2. The gods. Again, this has already been touched upon, but it's important enough to repeat. Especially since the Avatar Crisis (yawn), the gods are more involved in the affairs of mortals, particularly in attracting worshippers. A good character could send a prayer to Umberlee, an evil goddess who controls the sea, for a safe journey. Or Talona, the evil goddess of poison and disease, may get otherwise nice people to make offerings, in an appeal to stave off disease for their community.
3. Harpers. Chaotic good busybodies, but so much more. For instance, there's been a recent schism in the Harpers, where a breakaway group, called the Moonstars, I believe, has been started up in Waterdeep. Disagreements over methods and philosophies has caused tension in the ranks. As a result, my players' characters have run into at least one group of Harpers that are a bit more "on edge" than usual, quick to find fault with people's actions and being a little more paranoid.
4. The Zhentarim. My favorite organization. They are Lawful Evil, and their chief deity is Bane, the god of tyranny. I've made them PC enemies, sure, but sometimes, when chaos threatens to overtake a certain area, the Zhentarim sometime become unlikely allies. I've taken at least one "noble enemy" from the ranks of the Zhentarim. I've given them the philosophy of "Yes, we want to conquer the world, but if we let (crazed group of destructive chaotic evil types) have their way, there'll be nothing left to conquer. So, let's join forces, and we pledge (bear in mind, the Zhents as a rule ARE lawful) our co-operation until the threat's passed.
5. What lies beneath. The Underdark is a great place, and it's big enough that you can do stuff there with using the 'd' word.
Recommended Books:
Races of Faerun
Players Guide to Faerun
Magic of Faerun (it's a 3E book, as opposed to 3.5)
Lords of Darkness (evil organizations)