Before we can decide what the best weapon you should use against a zombie is, we need to make sure what type of zombie you are going to be encountering. The purpose of this survival guide and choice of weapons will be based on this scenario: Scientists and government agencies are working on a TOP SECRET project that has a cover story to help humanity and something inevitably goes wrong.
If you happened to have a practitioner of Voodoo out to get you, well you are on your own with that type of zombie. In that case maybe send the Priest/Priestess flowers.
Over the years, zombies have changed. While a lot of this may have to do with main stream entertainment and media, it is amazing how much truth you can actually find in a science fiction movie. In any case, it is always better to be safe than sorry. In today's world of zombies, they are usually created by some sort of virus or other blood born pathogen. Since you probably won't be doing IV drugs or having sex with these creatures, you will get infected by getting bit or getting the zombies bodily fluids into your bloodstream via a bite, scratch, or one of your mucus membranes (nose or eyes usually).
There are different types of zombies. You have the type that were already dead and have been reanimated. These are usually slow and easier to kill. The other type is an infected human. It can happen in a couple of ways. You can be bitten and change into the creature that has been given the name zombie or you could die from the infection (if you aren't torn apart as dinner for the hungry zombies) and then come back to life as a zombie. In this case, you can still be very fast and aggressive. You were only dead for a few minutes to a few hours.
Now that the brief history lesson is over, what is the best type of weapons to kill the undead? For many (myself included) firearms would be the 1st choice that would come to mind. While this would work, it is not always the best choice. Let's back track for just one minute. How do you actually kill something that has already died once?
For the most part, a zombie still has the "need to survive" part of the brain and the involuntary functions. For this reason you need to destroy the Medulla and the brain stem. A simple head shot may do the trick but most of the brain isn't actually doing it's job so a head shot through the lobe that controls speech may not take it down.
Now back to the firearms. They will usually work. You need to train as much as possible before you have zombies at your front door. Head shots are not as easy as they look in the movies. A rifle will work well from a distance is you happen to be on a roof top and have plenty of ammo. If you happen to be up close, a shot gun will be much more effective. At close range, a 12 gauge shot gun shell will remove all the brain so you are covered as to which lobe you need to hit. As any military person or hunter will tell you, you need a back up weapon. If you are a good shot and know your target, a small caliber handgun will work. A .22 in the right place will be just as effective as a larger caliber. I am an expert marksman and I think I would still be happier with something larger just in case, something like a 9mm, .40 cal, or a classic .45. Ammunition for all three of these pistols are easy to come by since they are popular with police, military, and the hobby shooters.
Now we come to a problem that isn't addressed nearly as often. Many zombies have the basic instincts. They can walk and they hunt (they seem to be opportunistic and do have some sense of smell. They will often react to sound as well as follow the herd. If you are popping off firearms like the 4th of July, you may attract more trouble than you want. Because of this, silent weapons will work better. A bow and arrow or crossbow (or even a spear gun) may work better. Another bonus to this is if you have only one or two zombies, you can dispatch them and then retrieve your ammo. Bullets can run out much quicker than you think, having ammunition that can be used over and over again is a big help.
If neither of these options is available, use what you have. Swords may seem like a good idea but unless you know how you use them, they may get you into more trouble. A run of the mill machete will work better. They are light, multipurpose, and easy to use. When you are being attacked, don't worry about the exact part of the brain you need to hit. Take off the entire head. Even if it doesn't re-kill the zombie, a headless zombie is much less of a threat. Plus, you can always crush the zombies head when it is off the body and make sure that the brain stem has been destroyed.
As a last resort, you can use an ax, club, or spear. The same rules apply as to the machete. If you can disable the zombie, it will be easier to kill.
That is a list of the best weapons to use against a zombie.
As a footnote, if video games and movies have taught us anything, DO NOT catch the zombie on fire until you are 100% sure it is not getting up. A flaming undead creature that wants to eat you is much worse than one minus the flames.