They're three sisters, their music sounds a lot like what might happen if one combined Vampire Weekend with Michael Jackson, or classic rock with 90's pop and R&B, they just released their first album on September 30th, and they're starting to get big. Plus, their bassist (on the right) makes what she calls Bass Face during performances (see below).
To start off: Try their Forever EP, if you're more interested in the Michael Jackson/90's pop and R&B elements of their sound. If you want something harder, try "My Song 5" off their album, Days Are Gone. For something else entirely, i.e. a sweet, stripped down love song, try "Honey & I" off of Days Are Gone.
Then listen to everything. I mean it. Everything.
This is a bit obvious, as Beirut actually opened for Vampire Weekend, but still. Their frontman Zach Condon (pictured above)-- or "Zach Unfortunate Last Name", as I affectionately call him-- formed Beirut after dropping out of school to travel Europe, an experience which undoubtedly shaped his sound. There are trumpets. Trombones and flugelhorns, even.
To start off: "Postcards From Italy". Literally everyone has heard it. Then I suggest The Flying Club Cup album. From there, you can make your own way-- it entirely depends on whether you enjoy the brassy sound and ukulele or not.
3) PAUL SIMON
Obviously this isn't a contemporary musician, but Paul Simon's influence is notable in Vampire Weekend's music. He's better known as Simon of Simon & Garfunkel, but his own music is a blend of folk rock with "borrowed" elements of reggae and African music.
To start off: "Diamonds on the Souls of Her Shoes" is probably his most popular song. It's off of the Graceland album, which is probably the best follow-up, and his most popular (and controversial, given all the "borrowing" of other music he did) album.
4) PETER GABRIEL
Also not recent but very much instrumental to Vampire Weekend's sound-- they even reference him in "Cape Cod Kwassa Kwassa". He was the frontman of prog band Genesis, which you may better know as That One Band Phil Collins Was In, but split off to do his own thing. The result is a blend of soft rock with Afrobeat and orchestral influences.
To start off: The Scratch My Back album features Peter's covers of a wide variety of artists, including Regina Spektor, David Bowie, Arcade Fire, Radiohead, and Bon Iver, so it may be a good place to start. His covers of "My Body is a Cage" and "Mirrorball" are particularly noteworthy. In my opinion.