It shouldn't be too hard to find a variety of weapons at the beginning of the game. Just go out and do some exploring.
Find a weapon you haven't killed with yet, kill a zombie with it, and then find another new weapon to kill with.
[THE FOLLOWING IS A LIST OF WEAPONS IN ORDER OF WHICH I FOUND AND USED THEM TO KILL]
1. Your Bare Hands
4. Diving Knife
9. Spade (Shovel)