Hi, frequent longtime lurker here.
I'd recommend the following:
The Avatar Trilogy: Starting with "Shadowdale"
http://search.barnesandnoble.com/Forgotten-Realms/Richard-Awlinson/e/9780880387309/?itm=6
This series forms the basis of most of the Lore in the Forgotten Realms. You basically
have to read it if you want to understand what occurred during the "Time of Troubles."
The Harpers Series: Any of them will do. "The Parched Sea" is the first.
http://search.barnesandnoble.com/Forgotten-Realms/Troy-Denning/e/9781560760672/?itm=9
They are standalone books, some with recurring characters, some not. They are excellent for getting an idea of the creator's minds concerning political turmoil and behind the scenes and events in the Forgotten Realms. They are also very refreshing in that the stories are from different characters' perspectives so you won't get a boilerplate type feeling from every single book. Because of the interconnectedness of some of the events mixed with the varying character's perspectives, you can get an almost "Thieve's World" type feel. (The much beloved collaborative effort edited by Asprin & Abbey, now resurrected in story form by Lynn Abbey after their falling out..)
Of particular note is a spinoff of the Harpers Series: "Elfshadow"
http://search.barnesandnoble.com/Forgotten-Realms/Elaine-Cunningham/e/9780786916603/?itm=1
IIRC, this details two characters in the Harpers series that readers especially enjoyed. It's well worth the read.
There are plenty of series and books based on the Forgotten Realms. Not all of them are of similar quality though. The two series above should give you enough reading material for a few weeks, at least.
I liked Salvatore's work until the characters became constructs that the story acted upon rather than characters who were set within a story. A subtle, yet significant difference of techniques. But, I suppose its inevitable with a series as revered as that: The characters are the only thing it can be certain that the audience enjoys so the story takes second-stage. The characters became fairly wooden after that and I lost interest in them. I prefer the unsullied ideals I formed of them early in the series to the latter, drawn out and exhausted archetypes.