This means armor, weapons and food – don’t go wandering into the wilderness without the correct equipment or, just as importantly, the best equipment. You wouldn’t traipse into the Nether with Leather armor now, would you? Always make sure you’ve got tip top gear and an ample armful of sustenance. Better to be safe, as the old adage says.
Set up Basecamp
A good thing to do when you find a cave you intend to mine is to install a 'basecamp' of sorts; a room equipped with a chest, crafting table, furnace etc. It will save time in the long run if you have a local spot to travel to rather than heading 'home' everytime you fill your inventory (which, considering the amount of cobblestone you may hammer through, can be quite often). When you get further into the game a network of Ender Chests will save even more time.
Never Dig Straight Down
You want to dig in a logical manner that will not make it difficult for you to return to high ground or to escape from danger. To this end it’s advisable that you find a cave, follow it to its conclusion, and then dig 'stairs' heading in diagonally down roughly one or two 'steps' at a time, about three or four squares tall and two or three squares wide. This will give you a safe set of stairs that you can branch from at any point you wish. It’s also a good idea to craft larger 'landing' areas as you go to necessitate easy expansion tunnels.
Bring a Light
Light is a wonderful thing so make sure you bring lots. Not only does it ensure you can see what you’re doing, but it also keeps monsters far at bay. Any pocket of darkness could work against you as a handy spawning point for nasty monster so don’t let this happen! Light up everything.
When underground there are all manner of unexpected scenarios that might play out. From getting ambushed by a zombie to breaking a wall and seeing a Creeper staring right back at you to encountering a rogue lava flow. As such mining in Minecraft is an endeavour that asks you to be ever vigilent and not to slack off. Be wary of your surroundings at all times and move as slowly as you need to, never assume you’re safe.
If you think there’s a risk in doing something then the best advise is often 'don’t do it'. Death in Minecraft is a crushing blow and you want to avoid it as often as possible. If it looks like something is going to take a concerted effort then return to basecamp, unload your accrued wares, consider what you’re doing, and attempt to overcome the issue while loaded only with the bare essentials.