So Zuckerberg just told a billion people to buy Player of Games. A-fucking-men! Player of Games is the obvious choice as a gateway novel for the Culture series (in fact, I always recommend it as such).
You can pretty much start anywhere with the Culture novels (though Look to Windward is a sequel of sorts to Consider Phlebas, Excession demands that you have a sense of how the Culture's AI's work, and Inversions assumes you know how the Contact/Special Circumstances sections of Culture society work). Player of Games takes the same theme that Banks always works with -- our protagonist is uncaring and uninvolved in the politics of the Culture and is thrust into a situation that'll decide the fate of an otherwise inconsequential world that the Culture has decided to uplift.
Banks had this wonderful personable approach. There's a bigger story at play but...it doesn't matter. The Culture is largely populated by nihilists (there's even a sect devoted to solipsism). Player of Games is about our protagonist, his AI handler, and the characters he's pitted against. He knows he has a mission to perform, and he knows the details and the consequences, but it's really simply his open and shut story. A day in the life of the Culture.