I’ll help you a bit:
The Trusty Pocket Knife
Known better as “Your Only Friend” once you decide to break from the shelter. Its clumsy butcher handling is superceeded by, well, everything you’ll want to accomplish in the very beginning. It’s tiny enough to lose a battle over multiple enemies, but surely will put down an occasional hostile entity. You can learn how to handle knives with this baby.
The Stick That Hurts More
So you’ve discovered Heavy Stick magic, and the variety it yields when you apply your crafts. One of the basics is the Pointy variety, your companion for cooking and poking small game from the very start. More than that, if you have excess weaponry on yourself you can really hurt an opponent closing in on you with throwing skill used. Remember, the more you craft from raw, rawer, rawest materials - the more you’ll know about even deadlier tools of war.
The Hammer of Zs
So you’ve picked up your broken heart for the decaying world and you’re heading for the first settlement only to find throngs of green, purple and other soda-coloured zombies walking about. You invite 'em to a choke point, crush their flesh-hungry faces only to find the Trusty Hammer on one of them. Don’t bother trying to poke your undead with screwdrivers, hacksaws or tongs of kinds, punch some holes with this baby that is similar to a crowbar you’ve (very much obligatory) made yourself, or at least it is when it comes to damage and wielding speeds. Keep it for building, too.
You’ll find that pre-Cataclysm world was a place of jolly sporting events - billiard tournaments, baseball games, swimming contests and, well of course, hockey team clashes. Now you can put the gear to somewhat different use, and aim it at the undead fans. The stick that pushes the puck is very well ballanced for bashing the skulls of the decayed, so you may want to stick with it for a while if you lack better options in current zombie infestation.
Cutting Implements Repository
Or the CIR, may be your undead neighbour’s home or, well, his rotting carcass. The survivors have more use of 'em, those sharp blades, and you may want to pack the Butcher or the Combat utensil of the Knife kind. You’ll be cutting everything in no time, from kevlar to sinew of the unlucky dog or wolf. You’ll stick with them through the obvious part of the game, and they’ll make a fine companion for zombie resistance until you find some larger and deadlier cutting weapons. Just don’t mind the steak knife or even worse, the butter knife’s dull blade, as the mentioned fence poorly in close encounters.
The Pain of a Crossbow
So you’re too into traps, aren’t you? Between the nasty metal jaws ment for bears and the double-barrelled trap for those crazy enough to ask for permadeath lies a Crossbow Trap. So you think, what the heck, I can do it - maybe you dodge it and then then BLAM! - you’re bleeding like a stuck pig, all because of a bolt fired from a powerful weapon, the Crossbow That Mangled You. Of course you wish to keep it, and learn how to make ammo and remember never again try and disarm a crossbow trap in the front of the weapon. You’ll find it satisfactory that you’re not the only one burnt by this powerfull projectile-hurling device once you get it on with some Zs. Needless to say, metal bolts can damage armor and cause electric pain to the CPU-brains of many robotic foes. Keep in mind when to reload, 'cause this baby has only one candy to set, draw and fire, at a time…
Have you ever swung an axe at something? Would you like Zs to get up and learn how to handle their weapons? Better dispose of those horrors and strip them buck naked, only to find they’ll never use the trusty Axe like you will instead of 'em. You’ll want this baby on you when dropping trees and such, but this also is a weapon of choice. And a deadly choice it is. If you don’t mind being set back by the slowness due to the size of this bone-crushing device, you can just wear as much protection as you can and go with choppity-chop-chop.
There are swords and bats, chains and barrel-rigged nail guns, but this is pretty much the spot.