I am under impression that either Guild Wars 2 is rubbish at introducing core concepts of the world, or I'm doing something horribly wrong.
For instance, during the A Light in the Darkness mission (chapter 6 of Personal Story), Trahearne mentions that, as a result of a botched fight with Elder Dragon Kralkatorrik, Snaff and Glint died; he then goes on to explain who Snaff was, never mentioning Glint again. The thing is, this is the very first time I heard of Glint in the game, and yet it's not even acknowledged who she was. (I know from accidental Wiki reading that she's supposedly a major character in the game's backstory.)
The same goes for several other things, like why is my character fighting Flame Legion (after fighting them all over Plains of Ashford, you get sort of enough information to piece together who they are if you explore the Black Citadel) or even a general idea of what Elder Dragons are (it would take me a good while to figure out that they're supposed to be six giant monsters and not, say, generic races of evil dragons, if I didn't read the wiki).
Have I done something out of the intended order, story-wise?
I'm playing a Charr; up to that point my major story decisions were: Iron Legion, traitor father, Euryale as the friend; joined Priory, and conducted Grawl research. I haven't done any dungeons. I tried to 100% complete every Charr area in order, before moving on to next; however, I was getting XP too fast to do them all before hitting the level cap, so I only managed to complete Plains of Ashford (at level ~22) and Diessa Plateau (at level ~43) before giving up that idea. I then did story mission, got 100% on Lornar's Pass, and then was doing a mix of story missions, crafting and whatever area I had to pass to reach the next story point, trying to do the story before hitting level cap and having XP burned.
Also, for what it's worth: currently I'm 2 levels away from Chapter 8 on that Charr character.
Best Answer
You should be able to play and enjoy most of the game without much prior background information - as long as you're aware that you are taking part in a world with a rich history and don't mind not getting a few references. This is probably how the majority of players proceed through the game.
For those who do want to get all the references and understand the lore of Tyria as completely as possible, you would have to go along the following steps and pay attention in cut scenes and dialogues (or at least substitute reading up on them in the wiki, which does a fairly good job of summarising them):
I guess the game developers wanted to strike a balance between offering GW aficionados a rich history of the world in which to play while allowing newcomers enough context to get going without forcing lore down their throat, and without forcing you to do everything in the "right" order. I understand how someone not familiar with the backstory might find some references confusing if they're the sort of person who wants to understand it all (as I suppose the OP does), but personally, I think they did a pretty good job of striking that balance.
How much of all this (which would take quite a large amount of time) one wants to do and how much one wants to skip is up to each player, of course.