Minecraft wiki entry for Hardcore_mode, associated text: Find the closest tree and punch it. Gather only 4 logs to save time.
Craft a crafting table.
Craft a wooden pickaxe.
Find the nearest stone and mine 9 blocks to start. If there is no stone nearby, dig an angle shaft into the ground to reach stone.
Craft a stone axe, stone pickaxe, stone shovel and stone sword.
Get about half a stack of logs.
Kill pigs, cows, and sheep to get at least 12 pieces of meat. Grab mushrooms if you see any. Having food now is worth the effort it takes to find more animals later.
Since it is safer to skip the nights early on, try to kill enough sheep to craft a bed.
If you do not have a bed to skip the night, do some branch mining under your shelter so you can get iron.
Start looking for a cave system, although it's best not to explore a cave or ravine this early on.
Don't make any stops. It is safer to wander around during the day with at least iron armor.