Minecraft wiki entry for Hardcore_mode, associated text: Sword. Use the best material available and discard damaged/weaker swords.
Axe/Shovel. An axe can shorten lumber trips and save daylight. One can be used as a weapon if your sword breaks. Shovels are unnecessary unless the player is doing lots of digging.
Bow. Conserve ammo and only use it on witches, skeletons and creepers. Skeleton arrows can be avoided by building a 2-block dirt pillar and hiding behind it.
Pickaxe. Indispensable in cave systems or the Nether. Use an iron or diamond pickaxe if available. Can be used as a weapon in a pinch.
Water bucket. They can extinguish you, prevent fall damage, let you safely travel down mountains or save you from an enderman.
Fire Resistance Potion. It can be a life-saver if you are mining or exploring near lava in the Nether.
Torches. Very handy when exploring cave systems. Place them on the right walls when going deeper so you can remember how to exit.
Dirt. Very common, can be broken without tools and does not obey gravity. Can be used to scale walls, build bridges, create quick shelters, stop skeleton arrows or prevent endermen from reaching you.
Steak. As plentiful and effective as cooked porkchops, but killing cows will also give you leather. Keep your hunger bar full whenever your health is not full. Golden apples can be used for dangerous situations.
Healing items. Items like Golden Apples and Enchanted Golden Apples, along with potions can save your life when fighting mobs or trying to swim out of lava.
Spyglass. Spyglasses allow you to zoom in on your surroundings, which can be very helpfuul for spotting a dangerous mob or a useful structure.
(Offhand) Shield. They can block Skeleton's arrows from pushing you and reduce damage from attacks. It is best to have it in your offhand so you are able to attack and defend easier than without.
(Offhand) Totem of Undying. They can help you cheat death. Extremely useful. With them, you can save your world.