The key thing you need to realize about Don't Starve is that every single item you pick up is useful. Before you start, it's important to press the "craft" button and look through everything you can possibly create. Craftable items range from survival tools to traps to weapons. Everything, no matter how seemingly insignificant (even seeds or blades of grass), could mean the difference between life and death.
Your first order of business is to identify what you can eat, which you can easily spot with a green icon at the bottom of the screen (likewise, anything you can craft will appear on the left side of the screen in green). As a general rule it's better to cook your food than eat it raw, so make a modest campfire (using basic resources at the start) and take in a few morsels every so often. Immediately try to gain access to the crock pot for the highest quality early-game food. But you don't have to eat everything, and certain foods or items can be used as fuel for your campsite.
That takes care of your hunger -- one of three major stats to manage in Don't Starve. The other two are sanity and health, which are a bit trickier. Once your sanity drops below a certain level, seemingly harmless enemies like fluffy bunnies become evil creatures, and you'll quickly lose out to a massive army of evil deranged, previously cute things. Increase your sanity by engaging in mindless tasks like picking up items, or wearing clothing. Of course, advanced players will eventually want to decrease their sanity from time to time to obtain special items. To go a bit mad, eat and pick evil flowers and use worm holes.
Health recovery is a bit difficult at first. Spider glands can be used for menial healing, but you'll want to resort to fully cooked food. Hunting is also a great option, and if you find "suspicious dirt," you can track down large animals, kill them, and cook their meat. Combat as a general rule is pretty simple, as you can get a few hits in on each creature before they "trade" one back -- so just dodge in intervals.
Pig Men are actually friendly until attacked, and can assist you in fending off unwanted encroachments (just don't feed them monster meat). Of course, you will eventually want to wander off into bigger and better pastures. Don't Starve would be fun enough if you just had to wander around a small area and stay alive, but as expected, there's more to it than that.
Now that you've gotten the basics down, it's time to understand what lets you survive for weeks on end. Firstly, don't use items you don't need for any given task. If it's not cold out, don't use earmuffs, and if it's not dark, don't use a torch. You also don't want to use non-weapons (like a work axe) on enemies -- stick to actual weapons for that, as they're far more effective and keep your gadgets sharp. Around Day Six you will need a solid camp, complete with a campfire, a stock of food and tools, and of course, storage.
Once you've acclimated yourself to surviving in terms of both offensive and defensive tactics, it's time to start looking into the Science tree, at which point you will begin your ascent into mid-game. When you're ready to settle down and start building an encampment or farm, you'll want to look towards placing structures. Structures can be a bit tricky because you'll need to craft the Science Machine first (and eventually the Alchemy Machine) to "unlock" the ability to build them (they're both under the science tab that has an "atom" icon). Think of it as "researching" a specific tree or tier in a real time strategy game. You'll also need the Science Machine to craft chests, which are paramount to your survival.
You also need to realize that each map you spawn into has its own ecosystem. If you go out and kill an entire colony of something -- it's gone. So if you make a huge camp near a specific resource or enemy and rely on that resource, you may have to venture further away to gather it and come back. Chop fully grown trees so that you can get better resources and plant more with pinecones.
Saving up earmuffs and other outfits comes in handy during the winter, and building the Winterometer will let you know, quite literally, when "winter is coming." As an aside, Wilson (the default character) grows a beard that allows you a little extra protection against winter -- so use him if you have trouble. (If you're curious as to how long it takes to unlock every character by way of XP, it's approximately 80 days.)
Another advanced tactic is to cage birds, then feed them dragon fruit to earn dragon seeds, which lets you plant more and more dragon fruit. When cooked, dragon fruit is one of the most effective food items in the game, restoring a large amount of both hunger and health. If you respect your environment, it will respect you back. That is, when it's not clawing your eyeballs out.
Lastly, Don't Starve is a sandbox-type game, so there's no real end to it. However, you can play the separate "Adventure Mode" gametype by finding "Maxwell's Door" (from the intro) in the sandbox portion. These stages have specific goals that are randomized, and you can't keep anything you have placed on the ground -- you can only bring four items, so make them count (Wilson's beard doesn't count, by the way).
Have fun, and stay sane!
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