As an interviewer from Pitchfork put it, James Mercer "started the Shins as a way of exploring three-minute pop songs with conventional chord structures." In "New Slang" that method of songwriting is very apparent. The song consists of some of the first chords any guitar player learns: A minor (referred to as Am from here on), C major, F major, and G major. But just because the chords are simple doesn't make the music uninteresting.
Take how the first verse, which refers more directly to regret and sadness, features the Am chord along with the other three, while the first two choruses that talk about how things could have been beautiful are made up of only G, C, and F. Each of the three chords in the chorus are major chords. What does that mean, exactly? Well, to be general about it, minor chords sound sad and major chords sound happy. It's interesting that the chords used seem to match the tone of what is being sung over them. Also, when the final chorus comes around and Mercer talks about a "good life [he] might be/doomed never to find," the "happy" chords he is playing can be seen to either reflect a sense of hope, or create irony.
If you want to hear how each of these chords sounds individually, go to this virtual piano, and click the button that says "Chord Mode." Here's how to make each of the chords:
Am: Click on A, C1, and E1.
C: Click on C, E, and F.
F: Click on F, A, and C1.
G: Click on G, B, and D1.
Press "Play Chord" after a chord is entered, and make sure only the right keys are pressed down.