One of the things I noticed right away when the world ended was how quiet it was. It seemed like the whole world was holding its breath, waiting to scream. You could hear things a long way away, and it was to the zombies advantage, sadly. I remember walking down a path and I thought my footsteps were too loud, even though chances were no one could hear them further than three feet away!
Once we got used to them, we realized how valuable they are and how much we can use them to save ourselves.
The one advantage we have over the zombies is eyesight. They can't see really well, mostly because the fluid in their eyeballs is clouding up with decaying matter, but they can find you by smell and sound, no worries there. But if a zombie is far enough away and you don't move suddenly or make noise, they might not know you're there. I've snuck up on a lot of zombies.
Use your hearing. Don't listen for the sounds of things that are supposed to be there, listen for what shouldn't. Dragging sounds have no place in a forest, nor do they belong in a store.
Use your nose. I've noticed once the crap filtered itself out of the air, my sense of smell has been getting better over time. Zombies smell, but not as bad as you might think. Sort out the smells that belong where you are and identify the ones that don't fit.
Trust your instincts. What people call their intuition is the subconscious mind sorting out the signals coming in through the senses and telling you exactly what is on the other side of that creepy door. Pay attention.