Zombies are so much fun work with in a story. There's a tremendous amount of material to draw inspiration from, but because the monster is an embodiment of something so primal, the fear of death, it's pretty open as to what exactly the zombies can and can't do in a story.
In my latest story, I opted to have the vast majority be your stumbling walkers. These guys will bumble about randomly unless they hear or smell you. Once then, their drive to eat your flesh overrides any other function in their animated brains. I also decided to have a little fun and include a couple of other types. There's Crawlers, those who lost one or more of their legs during the apocalypse, and then there's the Runners. The Runners were very fun to add because they ensure that my characters couldn't simply run across the university courtyard and be done with it. Nope. If they tried such a feat, the Runners would take off with a speed and drive that couldn't be matched even by their living counterparts.
Runners also serve as a reflection of my protagonist's fears. He's the first character I've written about who has an external physical disability that has made certain tasks like stairs and sports difficult for him. So, of course, he must do plenty of both in this story. Zombies in my University is a story about pushing to the real edge of your abilities and not letting excuses and fears hold you back in life.
You can follow the story of Sam, a college kid trying to escape his dorm room after the Zombies have ravaged his university, on Amazon or Barnes and Noble. Here's my cover: