At first, I decided not to play story mode but scavange. That way I have found some weapons and upgraded them, as well as build and upgraded buildings. You can do that as well, and if you are achievement hunter, you will have to do it anyway. If you wanna have smooth gameplay, this is what you need to do:

1. Upgrade hammer.
2. Upgrade basic pistol.
3. Upgrade SMG (optional)

With only those two (or three) weapons, you will have no problem beating whole game. Why? Because pistol can usually one shot zombies (close distance, from far away every gun is weaker).

You will get hammer very early in the game, if i remember correctly - in the second mission. Until then you will have to do with knife and basic pistol. Hammer can take down armored police zombies in two swings. Just hold your arm high above you head and swing twice, even on veteran difficulty he will go down.

Here you can see my examples how to easily take zombies down.










Rest of the weapons are optional.

One thing I noticed during veteran playthrough - my weapons were dissapearing. At one point I lost basic pistol, later my hammer and I am sure I haven't replaced them by accident. I don't know why, if it was a bug or not, but it may happened to you as well. In that case keep some usefull weapons on you (or maybe dont collect them, thus you wont get them replaced?). I just know bat may be glitched, it respawned in my inventory twice, so maybe it's good idea not to pick it up in the first place.