Any item you would need after the Zombie Apocalypse can easily be gotten. Just walk outside and get it.
Want transportation? Find any abandoned vehicle.
Want food? Go to any abandoned grocery store.
Want shelter? Either stay in your house or find an abandoned house or building.
Want medicine? Walk down to the nearest pharmacy or abandoned hospital. On your way there, stop by each abandoned house, walk inside and go through the medicine cabinets.
Of course on all of the above you will have to deal with gangs of roaming zombies who will try to eat you. This proves the point that the first priority will always be a weapon! Without a weapon you won't be able to get any of the other things you will need!
If you want a real life example of the closest thing we have had of a zombie apocalypse, look no further than the L.A. Riots of 1992. Fires raged, no electricity, gangs of roaming thugs looking to harm others, police couldn't help - they just drove as fast as they could through the area. The only people whose businesses remained relatively unscathed were the Korean store owners who stood guard with their weapons.
People who wish to survive a zombie apocalypse need to get over their hoplophobia
and start shooting zombies if they want to make it through alive.