The story of Wolverine in the Marvel universe is a weird one for new readers. Logan, the Wolverine, is dead, and has been for a while. His clone, Laura Kinney, also known as X-23, has taken up the mantle of Wolverine. And recently, a much older Logan has appeared, but he's not the one who died.
I highly recommend the series All-New Wolverine, which follows Laura Kinney as the new Wolverine. I am consistently surprised by the action, humor and heart of this book. It has made Laura one of my favorite characters in the current Marvel universe, and the biggest reason I was excited to see her brought to the screen in the incredible movie LOGAN.
Here is the All-New Wolverine in order, a great place to start:
However, I understand you may really want to read something with Logan. I would recommend one of these great self-contained books. Wolverine: Origin is as it sounds, the true story of Logan's earliest years, before the Weapon-X program. Death of Wolverine is, of course, the end of Logan in the current Marvel Universe. And Old Man Logan is an alternate-world story of a post-apocalyptic world where the villains won. That world's Logan has recently made his way to the main Mavel Universe, so you can continue his story here.
All three are fantastic self-contained stories featuring Logan, The Wolverine: