Borderlands' main storyline is as simple as it gets. There's an ancient vault that only opens every 200 years. The key to that vault is in three pieces. The pieces are held by bandits. Kill the bandits, get the key, open the vault. The real appeal of the game lies in its texture. There's a sense that the world of Pandora is gleefully psychopathic, and that your character is not so much a mass murderer who slaughters bandits and Crimson Lance soldiers by the hundreds as they are a simple contractor who is doing honest work in a dangerous environment. The tension between and humorous juxtaposition of these two ideas will be much more explicit in the sequel and the DLC, but there are already hints of it if accept the right sidequests and religiously read their descriptions.
It's interesting to me that of the game's major quest-givers - Dr Zed, Shep Sanders, Scooter, Lucky, Helena Pierce, and Patricia Tannis, half of them don't survive to the sequel. And the ones that do are the less serious characters, who all have a quirk that acts as a hook for jokes - Dr Zed's medical incompetence, Scooter's general redneckedness, and Tannis' horrifying psychological trauma. In a way, it feels like a mission statement for Borderlands 2 - the series is going to become more jokey and less serious.
Which, honestly, is probably for the best, because the "serious" parts of the game don't really work. Specifically, I'm thinking of the ending. By having the Destroyer come out of the vault, the game is trying to retroactively make the story about a hero chosen by destiny, someone who started off as an amoral mercenary, but who was thrust into events of cosmic importance. The problem is that this is sprung on you at the very last minute, with no sort of foreshadowing. Up until the Destroyer actually emerges, you are in fact an amoral mercenary, and the excitement of opening the vault is due entirely to the promise of alien technology, wealth, power, and women (I was really looking forward to meeting those vault women, too). To find out that the vault was really a not very interesting boss with no significant treasure just felt like a low blow.
But I'm not really saying anything that followers of the series don't already know (they even make a joke about it in the opening of Borderlands 2). Plus, they more than make up for it with the awesome treasure troves in the DLC.
Speaking of which, I'm on the fence about whether to play them. I definitely have to at least do the General Knoxx DLC, because it introduces Athena, who will be a playable character in the Pre-sequel. And all of them are fun enough to be intrinsically worth playing. However, level scaling is going to be even more of an issue than it was with the main game. I wound up being level 35 when the suggested mission level was 31. I think the bigger DLCs start at level 35, but I'm pretty sure that doing one will make the others unplayable. Ordinarily, what I'd do now is try to beeline through new game+, because the game automatically scales to your current level when you beat the Destroyer a second time, but I'm not sure I want to play this game for another 30 hours (especially since I have Borderlands 2 and all its incredible DLC waiting for me).
I think, what I'm going to do is try and rush through the General Knoxx storyline, while avoiding sidequests, in order to be as low a level as possible for Claptrap's Robot Revolution, and then, if I'm not too overpowered, try and squeeze in the Island of Dr Ned. I might play around a bit with Moxxi's Underdome, but honestly, that doesn't have much story interest, and since it scales with your character, I can save that for last (though I might pop in to use the storage for some of my cooler weapons).