This guide will only work for a maximum of 2 players, playing Classic, and the second player will take longer as the gears may take longer to find (or build, if you are inclined to do so). The reason for this is there is only enough fur for 2 backpack upgrades, and lots of item locations are changed in the new game mode.

Head South and slightly West to the mineral field. Hit a mineral stone until it breaks, take a stone, head to the nearest (generally directly South) tree, and destroy it for one wood. Immediately head back to base to make a tool: Scythe as all tools do the same amount of damage and only different unique uses. Use the map outside of the base to get the bonus objective if you are not the Identity Thief (as this roll already starts with the bonus). It does not matter what order you do this—get bonus objective then tool, or vice-versa—as it takes same amount of time either way.

After you have your bonus and tool, immediately head South West out from the base and switch to an empty slot for faster speed. You will run into a pack of 3 Moose. Kill 2 if you are a Traitor and your partner(s) are not very communicative or following you (following each other is generally not a great idea). If they are following you, kill the 3rd one and hopefully your partner will be quick enough to get the resources. If you are Innocent, you might as well kill all 3 because you have a (1-3)/8 odds the rest of the resources will be used beneficently. This is where the accuracy comes in. If you get good hits on the Moose, they will go down quickly and with as little damage to you as possible.

Now that the Moose are down, take a stack (3) of meat and the 2 hides provided by one moose. Then, go to the next kill and take the final hide to make a stack (3). Finally, head to the nearest bunker that requires 3 people to open. There should be 1-2 of around the base either North slightly West, North slightly East (above the base), or South slightly west. You may have run into the bunkers while fighting the Moose. Get help to open the bunkers. Seriously, get help: no need to be suspicious in the first five minutes of the game. With luck you will get the bunker open and it will have gears (Mechanical Parts).
Start gathering the gears. You will need at least 2 to get the backpack upgrade. Take 3 if you are a Traitor or if you are Innocent and the 1st objective is the Power Station and needs gears for repairs.

Hopefully you got your 2 gears. Head back to base and upgrade your backpack. This is the most important upgrade if you can get it as quickly as this guide suggests. You will have an extra slot for medical supplies to not bunch up the valuable space you have. There is no downside to making a backpack upgrade if it is done as soon as possible. The longer you have to wait to build this upgrade, the less effective it is as you could have been helping your team.

Now, backpack upgrade in hand, what to do with that raw meat? Pot Pies! Head out and get a stack (3) berries. There are berry bushes directly West of the cabin and more berry bushes North West of them. After getting the berries, get a stack or 2 of herbs (which is a total of 3 or 6). Afterwards head back to the base and make Medkit(s) and Pot Pies! Follow this order. You want to avoid dropping the crafted materials as much as you can to keep someone from taking the precious, PERMANENT buff Pot Pies* give (which stack up to 5 max).

*Note: A Pot Pie takes an entire slot: no stacking

Now this should have taken you about 5 minutes. Afterwards, you are free to go about doing the best for your team!