Minecraft wiki entry for Hardcore_mode, associated text: Never go unprepared into a cave in case you get surrounded or cornered. Always bring blocks, weapons, food, torches and armor with you.
Be careful around cave spiders. Poison cannot actually kill you but it will weaken you severely. Only fight cave spiders if you desperately need string or spider eyes. If you need spider drops it is best to fight plain spiders or loot dungeons if you have the choice.
Diamonds are not very hard to find if you travel in deep caves. They also have a 23% higher spawn rate. That being said, the common hazards are monsters, lava, and falls. The least you need is iron armor and a stone sword to go in. Diamonds are not difficult to find in Dripstone caves & Lush Caves. Watch out for mobs, they can spawn a lot there.
If you happen to fall in lava in the overworld, swim to the surface and place water above (not under) the lava. This way, the obsidian formed can provide you with an escape route.
Fall damage can be avoided by placing water to break the fall.
Do not ever dig straight down or up. Digging straight down could cause the player to fall into lava or a horde of monsters, and digging straight up could cause a falling block such as sand or gravel to fall and suffocate the player, or lava to pour down and burn the player. It is best to dig straight up or down in a wider pattern, to have a platform to stand on before digging down the blocks. Having a platform with a block above it could help allow the player to escape falling blocks without death.
Remember that a torch on the floor will break gravel and sand falling on it. This can save you from getting buried if you dig straight up (although you're still in trouble if you break into water or lava or a bunch of mobs).