For an authentic Ed Greenwood Realms vibe (the realms has since been redefined in flavour by many others, and unfortunately it seems that these days most realms fans are more interested in a sort of soap opera of the gods than anything else)...I'd suggest you try and incorporate some of the following:
Weird magic. Number one with a bullet...the Realms are hopping with weird magic...if not in quantity, then in quality. Don't be afraid to put in things like a grove where the trees float on little magical islands, or a cavern complex which has boulders continually rolling through it of their own accord. Likewise, nifty little semi-magical critters like flying daggers, along with crawling claws, lock lurkers and darkenbeasts are all very Realmsian in flavour. Likewise, spellbooks made out of customised materials and containing unique spells, with their own history, are just about as Forgotten Realmsy as you can get (the setting was first published in an article full of these called Pages From The Mages).
Nazi-like mages called Zhentarim. Authors and designers don't really use them as much as perhaps Greenwood intended. They should be up to no good somewhere nearby, at least, so let the PCs hear rumours of a Zhentarim skymage scouting the landscape on wyvernback, or Zhentarim mages hiring locals to clear out an old tomb, and then everyone involved except the Black Network members mysteriously die in an accident...that sort of thing. They're both secretive and active, and have resources to call upon, so PCs should be a bit paranoid about them ideally, and itching to confront them.
Spying bards and rangers called Harpers. At least one person that the PCs would never suspect of being a harper should be one, as well as perhaps some obvious candidates (such as the local bard or ranger). Like the Zhentarim, they're secretive (but with less resources) and will attempt to pull puppet strings on the PCs until they prove themselves trustworthy enough to know more, maybe even seeming to be the enemy. Make sure that most of the time they're continually absent, or have bigger fish to fry, or don't trust the PCs, if they're called on for aid. You can dangle membership with the Harpers as a bit of a carrot in front of players, as most good characters tend to like the idea of joining...then, for added entertainment, make circumstances force the players into conflict with them - even the Harpers get it wrong sometimes, or the PCs themselves might be tricked.
Uberpowerful, crazy, benign archmage denizens of forgotten dungeons. A senile archmage lich that think the PCs are students from the dungeon that was once his school of magic....the "back door" entry dungeon to the tower of a powerful archmage...an archlich who presides over an underground army of liches...a megadungeon with a crazy housekeeper archmage who might appear, cackle insanely and gate the PCs randomly. Ed Greenwood really likes the concept of the forgotten, crazy archmage, be he or she undead or alive, good or evil. If you make your own (they're very FR), it's good to make them mostly harmless, and probably bonkers, but obviously extremely powerful. Decorate their lairs with forgotten lore with the odd secret, arcane glowglobes, flying daggers, and other weird arcane paraphernalia that might date back to Netheril.
NPC adventurers in the Realms are romantic, and so are the sunsets. They tend towards jovialness and cameraderie in the face of danger, but many will quite happily face certain death three times before breakfast so long as they're with their friends. They're big on things like harping and storytelling around the campfire, which is a great way to introduce them, and seem to be very fond of Tymora as a patron deity, given their fatalism. Don't be afraid to cameo famous NPCs (Volo is a goodie; he turns up everywhere and doesn't have a stigma attached to him like certain others); that's what they're there for!
Don't forget the commoners. Greenwood's focus on sleepy little communities like Eveningstar and Shadowdale goes a long way to show that besides all the high magic and drama, the concerns of the local farmers, tradespeople and merchants aren't lost, and the PCs may find themselves searching for a missing pig for a local just as easy as trying to pry open a long forgotten dungeon. If you can keep mundane characters and their concerns in the game in at least a minor way, it provides contrast to the heroics and out-there stuff.
Anyway, that's the vibe that the Realms means to me from looking through Ed Greenwood's work in Dungeon and Dragon magazine, as well as the odd module. The classic places to set an Ed-style Realms campaign are Eveningstar, Waterdeep and Shadowdale....I'd suggest the former and the latter, simply because they're much easier to run.
