Baramos,
Start with High Violet (most accessible), go into the one-two punch of Alligator/Boxer, find their roots 'The National', see their growth in Sad Songs For Dirty Lovers/Cherry Tree EP... mess around with their singles and b-sides, then finish out with Trouble Will Find Me; which has a great encompassing feel of where they've been and where they're going.
They are defiantly a grower type of band but they're more than worth it. They just hit every mood and Matt's lyrics are so obtuse and in depth that they never get tired or boring, they always bring a new view through life onto them. I listen to them about twice as much as I listen to anything else. Heck my favorite track of theirs is even a b-side, their extra stuff even rivals their main... sort of like with Radiohead. They are that good...
Just this year alone, my Last.FM stats prove it, Top 10:
The National 1,940
Prince [solo, Revolution, NPG, 'love symbol', 3rdEyeGirl] 749
Sigur Ros 764
Brand New 523
Radiohead 488
Frightened Rabbit 470
Pearl Jam 436
The Replacements 362
Dave Matthews Band 290
Blink-182 280
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These guys are the best. Matt is one of the greatest writers of the modern music, the Dessner brothers have their fingers in so many awesome pies... and both Scott and Bryan are wonderful people who are the foundation to all of their songs.
Got into them right before Boxer dropped... Haven't looked back once. Have seen them live 5 times, even caught The Clogs twice. I love that they've released all of their ops in different colors. Bryce even signed my original boxer vinyl.
This past tour, when they were in St. Paul, Matt passed out his bottle of wine and we all shared in in the front row. Saw, Mistaken For Strangers finally a few weeks ago. It was remarkable, so funny... heartfelt, and honest.
These guys can't do no-wrong. Not only do they keep churning out fantastic records, they release fun-interesting singles (Portal 2, Win Win, etc.), they do crazy guest spots (Colbert Report) and unique songs out of the blue (Game Of Thrones, Bob's Burger's x2), have a great relationship with fans (LogicalHarm wrote a book about other fans live experiences with the band's approval, and she runs their forums) and the music world (Bon Iver, St. Vincent, Sharon Van Etten, Sufjan... etc.), and create artsy but fun documentaries about their journey.
I love TN.