So the dead are rising and shambling about. And for some reason they're really hungry. What do you need? Well food and water of course! But food and water are boring, so lets talk about armor, guns, and sharp pointy things instead.
Zombies are pretty strong, but not superhumanly strong. They have normal human strength, but they don't regulate the amount of force they use. Either they can't or they won't. And they're pretty uncoordinated and not especially bright. They're like big brain-eating toddlers. They're also stupid like toddlers, so about all they can do is shuffle up to you, grab on with all their might, and bite down really hard.
The good thing about all that is that defenses are pretty easy. A good concrete wall or a strong reinforced fence will hold small numbers back. But a larger mob can be more problematic because a bunch of people can exert a surprising amount of force. Most walls and fences are really only part of a layered defense.
What about body armor? Well you don't need heavy bullet proof vests or kevlar helmets that's for sure. Not to protect you from zombies at least. All you need is something that they can't bite or tear through easily. A set of full motorcycle race leathers would probably do it. Add a set of boots you can walk comfortably in for long distances, a good hockey or football helmet, and something to protect your neck.
If the dead walking is a supernatural event, you could stop there. But there is a chance the zombies are because of an ancient plague or modern genetic engineering. If so, these things only seem to be spread by a zombie's bodily fluids. So pack safety glasses and a surgical mask or bandanna in case you get some of him on you when you kill him. And maybe some cleaning chemicals for afterwards.
I see a lot of people saying 12 gauge here. Or machine guns. No, not really. While big guns might be useful if the living dead cause an overall breakdown in the social order, they're not necessarily that useful on zombies. Why? Because the only shot that counts on a zombie is a shot to the head that penetrates the skull and scrambles its brains. Shotguns and machineguns deliver a cone of fire, but they aren't going to give you an accurate head shot over long ranges. Plus the ammo is expensive and gets really heavy, really fast.
My suggestion? A decent .22lr rifle and a lot of ammo for it. The gun itself is cheap to buy and most are accurate. Ammo is cheap so you can stock up. Ammo is light weight so you carry a lot. Now you can own any zombie that gets within about 25 yards of you. They're also quiet so your gunfire will be less likely to attract other zombies who want to eat you. They're even quieter if you shoot from inside a building or with a silencer (which is unfortunately illegal to own in my state).
If you want more lethal range or something that will deter people, get a .223 rifle as well. The AR-15 is a great choice because they're generally quite accurate. This will let you own zombies over greater distances, but the gun is really loud. Really really loud. Just start making blood curdling zombie groans yourself loud. Which will attract more zombies to your position. So you have to fire and move or prepare to significantly reduce the entire local zombie population.
As always, handguns suck over everything but short distances. But at very short ranges they could be all that saves your life.
Manual Brain Scrambling
Unfortunately, you might run out of ammo. Well crap, now what? Well you need a hand held and muscle powered tool that will let you destroy a zombies brain or decapitate him. I prefer the former because Zombie heads can still bite and they tend to wind up under foot.
Don't use baseball bats. Hitting two round objects together, like a bat and a skull, is a recipe for ineffectual glancing blows. Plus they break easy. A friend of mine once broke two aluminum bats in the space of 15 minutes. Yes he was a burly blonde mountain of a man, but do you want to risk it if the dead are walking the earth?
A good makeshift solution is to buy an ax, sledge hammer, or crowbar from the hardware store. Swing it over your head and into Zeds. These have fairly long handles which keep you out of reach and can do a good job of cleaving or crushing a skull. And they're cheap.
If you want to be flashy, more specialized weapons are always a possibility. Max Brooks likes the shaolin spade, but its a decapitation weapon and where do you get one? I like the good ole poleaxe. Spike him through the eye or bring it down on his head. A flanged mace isn't a bad idea either. Spears are cheap, but I don't know how easy it is to penetrate the skull with one. If you're looking at swords, I recently discovered Windlass Steelcraft's Civil War reproduction sabers. They're sturdy, reasonably well balanced, and often priced under $100. That's a great price for a sturdy sword with a decent temper that will take a good edge. Or go buy a katana like everyone else.
There are of course other options for killing zombies. Just keep in mind that if it doesn't destroy the brain, it doesn't do the job. Unless you rain fragments down on their heads, blowing them up doesn't work. Don't unleash a flamethrower onto the zombie hordes either. You just get zombie hordes that set things on fire.