What sourcebooks should I get to make a fully-themed Forgotten Realms character in 3.5e?
Looking for setting information on:
Step 0 is to ask the DM what's allowed in the campaign. The Forgotten Realms is a vast setting, and usually anything you can imagine ends up there eventually somehow. Seriously, if you picked a D&D 3.X book at random, chances are there's something (and, more than likely, a lot of things) in there for the Realms or that the DM could put in the Realms and it wouldn't be out of place.
If the DM's following the broader reading of the primary source rule, likely the DM will rule that primary source for Forgotten Realms setting information is not the Forgotten Realms Campaign Setting but the Player's Guide to Faerûn that updates much of the Campaign Setting in light of the 3.5 revision. That doesn't make the campaign setting obsolete—the background information is vast and, really, if nothing else, the book is beautiful.
If you want an overview for the Realms, you get…
Then you can get a region-specific sourcebook or two, like…
(So you know, regional feats are extremely complicated mechanically by frequent revisions. It's best either ask the DM what region your PC must be from to take a particular regional feat or pick your PC's region and ask the DM what regional feats are available. Trying to make heads-or-tails of what and what's not a regional feat for your PC's region using only the books is a test requiring vast endurance and patience.)
As a player, you'll likely find unnecessary Anauroch: The Empire of the Shade, City of the Spider Queen, Cormyr: The Tearing of the Weave, Dragons of Faerûn, Expedition to Undermountain, Into the Dragon's Lair, Lords of Darkness, Lost Empires of Faerûn, Monsters of Faerûn, Pool of Radiance: Attack on Myth Drannor, Serpent Kingdoms, Shadowdale: The Scouring of the Land, Sons of Gruumsh, and Twilight Tomb. These are either adventures or monster books or unrelated to the workings of most PCs.
Other books also contain information specific to the Realms. For example, Complete Warrior, the Dungeon Master's Guide, Epic Level Handbook, Fiend Folio, Monster Manual III, Monster Manual IV, Monster Manual V, and Weapons of Legacy make mention the Realms to some degree or another.
Note that this list is really basic, and even some weird, ancillary books might have interesting material applicable to your PC. For example, you gotta dig deep to find the absurd magic weapon special ability smoking (Lords of Darkness 180) (+1 bonus; 0 lbs.).
These I remember being really helpful.