As many of you know, hardcore mode isn't the easiest game mode to play. Most of us would just hide in our wooden houses at night till will we muster up the courage to go out and hunt. Not many of us make further than a dozen days. Exploring is risky because you may loose your home where all your valuables are kept, mining is too risky because you may stumble upon lava, and even hunting passive mobs can be risky.  So here are tasks for your 5 1st days you must prioritize if you want to ensure your survival.

Day 1:
On day 1 just gather up wood, preferably till you have 2 stacks and get at least 8 cobble stones. You already have the main ingredients to make a furnace, torches, chests, wooden planks, crafting benches wooden tools and so on.
You should have a lit up house by nightfall. If you saw sheep before then, murder three for their wool. If you did this then you can skip the first night. But if you didn't then barricade your home entrance with any block because Zombies in Hard or Hardcore mode can break wooden doors down.

Day 2:
If you’re unfortunate and didn't receive any apples while punching trees down then you've got to hunt passive mobs for food. Once the sun comes up scout the area for any Creepers, these do not become friendly and will not burn down under the sun. Avoid the areas where hostiles are and stack up on enough food to last you a long while. Collect seeds and leave markers to your home while doing this. Before sunset you should already be at home cooking food. By now you should have wool and have crafted a bed.

Day 3:
Hunger should no longer be a problem for a while. For the time being there is no need to go outside, dig up stairs going down and light it up with torches when it gets too dark. Collect iron, cobblestone and coal while doing this and once you think you've gone deep enough begin digging out a 9x9 square at least 4 blocks tall.

Day 4:
Return to the outside world to dig up some dirt (at least 80) and collect the same amount of seeds. If you have 3 iron ingots craft a bucket and get some water. With these and the 9x9x4 room you dug up underground you can create the most efficient wheat farm Minecraft has to offer. Mine the stone on the floor of the room and replace it with dirt but leave the center of it empty, that's where you're going to place the water. Use a hoe to tilt the dirt and plant the seeds. Water will hydrate tilted blocks up to 4 blocks away from it so you don't need to worry about your seeds going to waste. Provided you lit up the room well the seeds will grow into wheat in 2 days time.

Day 5:
Now that your food supply is guaranteed you can repeat Day 3 or enter a cavern. Either way you should stack up on iron and make swords and armor out of it. After this step I can no longer help you. You can either keep mining as long as you keep lighting the mine up avoiding the surface world or explore the surface.


 
Even I'm confused with what to do with my Diamonds, should I make a sword and Battle mobs at night or should I make a pickaxe and obtain a diamond pickaxe so that I can mine Obsidian to go to the Nether...?