Well, the good news is that the war is small enough that it's reasonably easy to grasp.

Admiral Woodward, Task Force Commander wrote a pretty easy to read book named One Hundred Days, which is a very good startpoint. "Sharky" Ward similar wrote "Harriers over the Falklands" to give a more air-focused account, though I think he's a little self-aggrandising.

Martin Middlebrook wrote two books from the historian's perspective: The Falklands War, and The Argentine Fight for the Falklands, which looks at it from the Argentinian point of view.

There's plenty more, but those three ought to be a great start (Sharky's book a distant fourth)