Are there any important things I should and/or shouldn't do?
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Don't compare it with BioShock.
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It's a game with an emphasis on curiosity and the unfamiliar. Just play it and don't worry if you're doing it "wrong."
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Conserve ammo. Go for headshots, or use stealth to get past or get up close with a knife. Also, always check for traps with the wires, even when you're being attacked and are falling back to some place.
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Pay attention to story bits amd conversations. They lead to the secret ending.
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Find a volt driver. Charge it up and it makes the scary monsters go away. Explore all of the things and you'll find one pretty early on in the game.
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Don't run into a fight Call of Duty style and expect to come out alive.
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When in enemy areas with large populations, specifically people, stealth is your friend. Learn how to use throwing knives, and always, ALWAYS aim for the back. The good thing is, you can retrieve knives from enemies after you've used them.
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Scavenge the hell out of the areas you're in; any ammo you find will be hugely helpful in your save game. Try not to use military ammunition against enemies on lower difficulties, because it's a waste.
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when you get to armory station about halfway through the game, make sure you buy a stealth suit, or combat suit (depending on your play style). It is very easy to not notice that they're there, and you'll regret it if you miss them.
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What everyone else said. Also, look out for the Librarians and the shadows.
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Buy air filters, after around the middle of the game you will never be able to purchase them again.
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Save your ammo. If you're stuck in a situation with a shitty gun or no ammo you're screwed.