My new(ish) control wizard alt forced me to remind myself of where I could get some free bags for her. You can get three for each character, and I was surprised that I didn't find a clear guide on how to get them.
The first is the Adventurer's Satchel you start with, which has thirty bag slots.
The second is the Adventurer's Knapsack with eighteen slots, which you get from completing the quest "Close to the Crown" in Blacklake District. I think you also have to complete the quest you get from Sergeant Knox immediately after coming from the tutorial on Sleeping Dragon Bridge first, the one to check on the Crown of Neverwinter. That story then continues via Private Hathidon at the entrance to Blacklake District: "Seeking a Suspect" leads to "Hot on the Trail", which then leads to "Close to the Crown".
The third bag is the Adventurer's Pack with twelve slots, and requires you to complete most - though not all - of the quests in Neverwinter Graveyard. The key quest is called "Wheels Within Wheels" and is given by Harper Windle. To unlock it, you first have to do a bunch of others from the other NPCs. I believe that all of these are required, but it's possible that I did one or two that weren't because many of them were close together and I didn't want to have to run back every time just to test if that one quest alone would unlock more.
From Doomguide Volahk: "The Missing Doomguides" and "The Dead and the Dying". "A Doomguide's Duty" is not required, however it comes immediately after one of the required quests, doesn't take long and rewards a profession back, so it's also worth doing.
From Doomguide Orran: "Rest in Peace", "Thinning the Horde" and "Red Rituals".
From Christopher Chettlebell: "Correcting History", "Dragon Attack", "History Lessons", "Opening the Way" and "Clockwork Guild Tomb".
From Captain Zemmer: "Mercenary Nature"
From Harper Windle: "Enter the Dragons", "Artifact Snatch" and finally "Wheels Within Wheels".