To know your enemy is to know how to fight him. Zombie fighting won't come until we've achieved the first levels of the Hierarchy of Needs. Meaning safety and shelter. To get to safety and shelter you have to survive the first panic of the zombie invasion where it's every man for himself.
The zombie attack is on, what do you do?
First it will help to know what kind of zombies you're up against. There are two types. The Living Zombie, a person who is still alive but feels no pain and is driven to rip you to shreds. This living zombie is usually from a chemical spill or a virus or a meteorite landing near by. The second type is the animated, or classic Undead Zombie. This guy has just risen from the grave and is being animated by an outside force. They can be arisen by cosmic rays or magic.
To tell them apart may not be all that easy at first. Both will present messed up. The living zombies tend to eat their own lips off and will look like an undead zombie. Living Zombies usually don't smell as bad and move more easily. Undead Zombies require effort to move, are slow with slight bursts of speed and smell dead.
The Living Zombies are usually limited in number. They are fast. They can sprint until their hearts explode and then a little more. Since they are alive they need things like heart and lung function. Sadly, they will run you down. You're not going to out run a Living Zombie. They will catch you. Luckily, you don't have to out run them, just the other guy trying to out run them. They are also very strong so hand to hand isn't going to work. But, they need things like eye sight so a camera flash will daze and confuse them. Also gunshots to the chest to take out the lung and heart will also be to your advantage. Just keep in mind even a heart shot may take them several seconds to bleed out. Seconds they can still attack you. Shots to the stomach isn't going to bother them at all. Put two quick shots to the chest and get outta there.
Let's talk about 'Head Shot' for a moment. Head shots always work, but who can hit a random, shifting target that's smaller than a pie plate? Shots to the cheek don't count, you gotta hit them in the brain. Good luck hitting that! With Living Zombies a deer slug to the pelvis will cripple them and may cause them to bleed out. Rather a crawling zombie than a running zombie. I can run from a crawling zombie. Living Zombies burn nice once you get them going but you're not going to hang around for that. Get going!
Undead Zombies are slow. Run from them. That was easy. If you're trapped and have to fight, here's some tips. Undeads don't need to breathe or have heart functions so shots to the body are a waste of time. Even a deer slug to the pelvis will only punch a nickel sized hole in a rotting bone and not slow them. They don't burn as well and usually there are millions of them. These are the guys that will quickly overwhelm you. The only way to take them down is to destroy the brain. Shotgun buckshot at close range but this puts you at close range and we're also hoping you have a shotgun. Practically, you need to make a weapon from what you have in your current environment. Cowbar? 2x4? Here, I suggest a blunt object. Bats are nice. Aluminum is best. Aluminum won't break like wood, but grab what's handy. Undead's are usually calcium deficient, so crushing their bones is easy. Get a garbage can lid for a shield. Attack what's your first threat, his out reaching, grabbing for you, arms. Keep the zombie from grabbing you with the shield and swing at the outreached arm, shattering the bone. He needs bones. A zombie who can't grab me is to my advantage. Now swing the second shot at the head, aim for the top of the ear and follow through. You have to do enough damage to turn that melon into mush. In the fracas, look for strikes of opportunity. Can't reach the head? Swing for the hip and break that bone. A zombie who can't walk is a zombie who can't chase me.
Now run. Don't dawdle trying to 'finish him'. You're surviving, not sightseeing. Get outta there.