I think, if I am expecting to be on the move a lot... Then the 22LR weapons are pretty smart. Anyone who has ever carried much ammo at once knows that high powered rounds are quite heavy.
22LR has the advantage of being small and light. Should be quite effective on zombies. After all, from watching The Walking Dead, we have learned that zombie skulls apparently soften enough that pocket knives penetrate fairly easily. A 22LR Hollow Point should be quite effective. Plus, 22LR is pretty easy to suppress if you're worried about attracting more zombies with the noise.
So... I'll carry my Ruger 22/45 pistol and my Ruger 10/22 which I have scoped. A back pack of ammo would be a crap load, and the ammo will work in either weapon.
Then I'm thinking a good fighting knife like a K-Bar and maybe a katana designed for cutting (not a wall hangar).