My dear, you're asking the impossible.
That adventure takes place in a setting called Forgotten Realms, in a particular region called the Sword Coast. In the Sword Coast alone there are far, far too many important NPCs to name. All you need is to know the NPCs in your adventure.
That said, you can learn quite a lot about the region in this:
https://www.amazon.com/Sword-Coast-Adventurers-Guide-Accessory/dp/0786965800
FR is a setting that's been in the works since the mid-60s. It's impossibly huge. Do not feel the need to know much about it.
I suggest just reading up on areas you'll be adventuring in and ignore the rest. HotDQ and many other 5E adventures take place on the Sword Coast. That's still giant, but at least narrows down the amount of stuff you want to know. There's a published guide to it for 5E as well:
https://www.amazon.com/Sword-Coast-Adventurers-Guide-Accessory/dp/0786965800
Do you have the Sword Coast Adventurer's Guide? It has a couple of additional backgrounds, one of them being the Faction Agent (pg. 147).
Basically she's a member of the Zhentarim, or the Harpers, or the Order of the Gauntlet, et cetera. But she obviously doesn't remember. I'd have certain people recognize her, and help her specifically (not the other PCs). At some point, give them a quest working against this organization, but let her discover at a crucial moment that she's a member. And that the organization actually has a real good reason for doing the things they did so far.
Some of those factions have a code book, and an emblem/medallion/thingy for identification. So you could give these items to her, as small clues/hooks to let things make more sense slowly.
The "canon" world of DnD includes many things. Many planes of existence, many realms, and so on. Waterdeep and Baldur's Gate are cities along the Swordcoast on the continent of Faerun in the "Forgotten Realms". They are only one small part of a much larger DnD universe but is where most of the 5e materials are located.
The "history" has been added to over countless out of game years, with many authors and through many mediums, including a lot of novels. There are centuries of in game lore. It is fractal and never-ending and could drive you nuts trying to learn it all. It isn't necessary.
Unless you really just LOVE studying lore or you intend to take the AP History Exam of the Forgotten Realms, I would skim through various resources, read some short summaries of current lore and make it work for you and your table. Ditch what doesn't work, keep what seems interesting and go from there. Most will never be needed at your table.
There are some great guides that have been updated to "current" times for both Waterdeep and Baldur's Gate. You might hop on the DMGuild website and do a search for those cities in particular.
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Here are some more in depth resources for the broader picture as well if you really want to delve broad and deep, however....
Forgotten Realms Guide to Faerun Including the Sword Coast
Updated current info on the Sword Coast: The Sword Coast Adventurer's Guide
You can get it for less at Amazon or Jet(about $28 before coupons/discounts).
The Sword Coast Adventurer's Guide (SCAG, for short) is rad. Brief histories of the Sword Coast and other parts of Faerun by region. Profiles on the various races and classes. Couple new class archetypes and a few new spells.
Amazon has the SCAG on sale now for 31$.
I also recommend Matt Forbeck's DUNGEONOLOGY book. It has a mini-setting manual on the Forgotten Realms (FR) but also has the best map of the Sword Coast in print to date.
You can also easily ignore the mechanics of previous editions and just focus on the flavor text in the campaign setting guides.
If you can find the 3e or 4e FR Setting Guides on eBay or Half Price Books for cheap, they are pretty useful.
The Volo's Guide to series from AD&D are boss as well; V'sG: to the North, to the Dalelands, to the Sword Coast, etc. They should all be on the DMsGuild in PDF for dirt cheap.
When all else fails, look for FR setting books written by Ed Greenwood (Volo himself); Ed is the Tolkien of the Forgotten Realms.
It's in this, released just a couple of months ago. Enjoy!
https://www.amazon.com/Sword-Coast-Adventurers-Guide-Accessory/dp/0786965800
I don't think they've ever published a guide to the whole world, but let's be real, no one really cares about the rest of it.
The update to the 5e lore starts with The Sundering Series. Despite its name, that series only covers a narrow fraction of the actual event, tho. It is followed by Spellstorm and Death Masks, by Ed Greenwood (updates on Waterdeep, lots of misc. lore scattered throughout the book about returned deities like Eilistraee, the Weave, etc...); and by Fire in the Blood, Ashes of Tyrants and The Devil You Know, by Erin Evans (they include updates on Azuth, Mulhorand, Unther, the Dragonborn, Asmodeus, and Tieflings--you may want to to read the rest of her series first, tho). Keep in mind that even with those books, a lot of the changes happened offscreen, and their explanation was often left as a "mystery" or handwaved as "Ao's work". The book to read to know the status quo as of the 1490s DR is the "Sword Coast Adventurer's Guide".
If you want to read the lore that brought to 4e (Spellplague), I'm afraid that in tis case too there isn't much about the Spellplague itself. The Empyrean Odyssey and Abolethic Sovereignity trilogies are all that there is, and they don't even focus on the Spellplague itself. The Haunted Lands trilogy is also pretty cool. Keep in mind that all that you'll read in those books (except the in the Haunted Lands) no longer holds true in the current Realms. Like in 5e, a lot of changes happened offscreen, and the 4e FRCS is the book to read to know the status quo in 1479 DR.
Salvatore also kept writing Drizzt, but his story is mostly self-contained and never touches anything beyond Drizztland. If you want a reading order for his books, tell me and I'll provide it.
Forgotten Realms, hands down. This is great, if you're playing 5E:
https://www.amazon.com/Sword-Coast-Adventurers-Guide-Accessory/dp/0786965800
The SCAG has lots of good stuff, even though it doesn't cover all of FR.