A long melee weapon
As the old adage goes, "speak softly and carry a big stick." In this case, we're speaking softly to avoid getting the zekes to pay attention to us, and we're carrying a weapon long enough that we can kill zombies without being within their reach. Although we generally associate zombies with biting you in order to infect you, the reality is that you can catch their disease from something as simple as a scratch or getting some of their zombie saliva in an already open wound. Pretty much every bodily fluid that a zombie has left carries the infectious strain, and you just don't want that anywhere near you. The longer your weapon, the less likely you are to be accidentally scratched in the fray. Sharp weapons are preferable because it's more efficient than blunt. Bashing in a zombies brains may be fun, but it takes a lot more than a few swings, and can be really inefficient if you're taking on more than a couple. See if you can find a spear, a farmer's scythe, or a similar pole arm type weapon, or make one yourself.