I'm not a professional historian, and I'd like to read more about the Stasi than what's on Wikipedia. I found these books (which happen to be conveniently available in Kindle editions):
Would any of the above make a good starting point, and if so, which one should be first?
Stasiland is ok, but not particularly fact based, it expresses the 'tone' of the East German State in memory very well. The other books are indeed a good place to begin, though I would recommend 'Anatomy of a dictatorship' by Mary Fullbrook as a preliminary read to better understand the context of the Stasi within the DDR.
As mentioned my /u/ROBOTNIXONSHEAD, Stasliand is more of a narrative and Mary Fullbrook's book is indeed very useful.
A few more useful books I can suggest are:
Within Walls: Private Life In The German Democratic Republic, by Paul Betts
Behind the Berlin Wall: East Germany and the Frontiers of Power, by Patrick Major
Seduced by Secrets: Inside the Stasi's Spy-Tech World, by Kristie Macrakis (Looks more at the technology used)
And for a good overview of Soviet policing in general - this book is great! : Policing Soviet Society: The Evolution of State Control, by Louise Shelley (All available on Amazon)
My dissertation at uni looked specifically at the Stasi so if you're unsure about anything, feel free to drop me a pm.