In order of priority (feel free to edit):
Critical
1) Fire axe, or Hatchet: With the fire axe being a priority over the hatchet. Back away from the zombie until it does its swing animation and then smack it with the axe.
2) Can Opener: Combat Knife as well, both do not lose you food. But the Combat Knife seems to be pretty rare. If you use anything else, you'll lose about 40% of the food from a can. A kitchen knife is not a combat knife.
3) Bandages: Made by ripping up a shirt only. You can also find bandages in medical kits.
4) Compass: Third priority until you memorize the map. Not that important if you're rolling with a group.
5) Canned Food: Beans first, followed by tuna, sardines, spaghetti and so on. Stay away from rice, cereal, and powdered milk. Rotten food is a no go.
Important
6) Any Jacket: Hoodies, Raincoats, TTKso clothing. Helps with the condensation blowing out of your mouth plus give you extra space.
7) Backpack: The bigger the better, preferably green. The mountain backpack having the most storage (35 slots). Although I'm going with the Hunting backpack (30 slots) do to the OD green drab color.
8) Water Bottles or Canteens.
9) Rifle/Gun. Really a mater of preference. Which ever rifle you get, you want a scope to go with it, Pristine or Worn. On the M4, the hand guards should be pristine to help with accuracy.
Everything else
10) PASGT Helmet.
11) Pistol or Magnum Revolver.
12) Extra ammo.
Note I didn't say canteens or water bottles as critical. These are important, but not critical enough compared to the above items. If you work from well to well, or pond to pond and drink until your guy can't drink anymore, water won't be an issue. Water will become an issue when they add infections in. Supposedly drinking from a pond carries a risk of infection. However the infrastructure (Tin can, fire places) aren't in the game yet, so for now you can drink from a pond just fine.
Medical Supplies
Aside from Painkillers and the Morphine Auto-injector, I really wouldn't worry about these until you get in with a group of players. Most of them (like epi-pens, saline bags and blood transfusions) are really designed for another player to apply them to you. The Morphine Auto-injector heals broken bones. If you've lost a lot of blood and are greyed, simply eat and drink until your character can't anymore (do not eat beyond the warning as you will throw up and make maters worse). Simply log off for a few hours, or keep playing and gradually your blood levels will go back to normal. You do not need the "healthy" status to regen blood. Having the healthy status makes the process faster however.
Blood test kit: If you don't know your blood type, use this kit as soon as you find it and jot your blood type down. Then consult the
Blood Compatibility Chartto determine what blood can be donated to you. To be honest, as long as you eat and drink plenty (while greyed) the blood will return. So that whole process is currently a bit trivial. It will definitely become more important when the game matures a bit more. Part of the problem is that someone else has to give you the blood transfusion. So it's really designed for squad and group play. O- can donate to anyone. One of the things I'm wondering is if we'll be able to label blood bags...