Most tutorials are about what to do. This is about what not to do.
Sorted (loosely) by ascending order of obviousness.
Don't dig straight down
This one is fairly straightforward. If you mine the block you're standing on, you're quite likely to fall straight into a cave system and take falling damage/be killed by mobs, dig into lava and lose all your items, fall into an underground lake and drown, or fall into a dungeon and be massacred. Notch has even added a splash text to this effect.
Don't dig straight up
This isn't quite so obvious. However, if you mine the block above of you, you can release all sorts of nasty things directly on top of you. Gravel, sand, water, lava can all kill you by suffocation, drowning, or burning. Placing torches (or ladders if you have any) underneath where you dig up can prevent suffocation, you may still drown or burn.
Don't try to kill creepers on your first night
Creepers aren't quite like other monsters, as they explode when they get close to you. This means that you should get a Bow and kill them from a distance. Only attack them if there is no other choice. If you do, try hit-and-run tactics on them.
Don't mine valuable blocks with low level equipment
If you don't have a Iron or Diamond pickaxe, Redstone, Diamond, and Obsidian won't drop their valuable contents. Instead, they'll just be destroyed. A good indication is the amount of time it takes to appear to be destroyed. If you have been mining a block for more than 10 seconds (with the exception of obsidian, for which you should wait ~30), it is unlikely to drop resources when destroyed.
Do not play with fire
This is a no brainer. Fire and Wood results in more fire! Although as of Beta 1.3, forest fires no longer burn indefinitely, they no longer leave any wood standing. This can remove entire forests, as well as accidentally burning down your house, entirely, with none more wood standing. As Smokey the bear says: "Only you can prevent forest fires."
Do not build a flammable house
Planks, logs, leaves (trees), wool, and bookcases are all highly flammable and susceptible to explosion damage. All will be blown up by a creeper mercilessly, which would leave you with no protection from mobs. Cobblestone is a cheap, abundant alternative. Dirt may also be used, but only as a last-ditch effort, as it is very susceptible to explosions.
Try to avoid using primitive tools
Planks have better uses than tool crafting, and should only be used as such when there is no alternative or when you are just starting. The same applies for cobblestone, and then iron. As you progress down this line towards the final option, diamond, you should find enough of each resource to create the specific type of tool. You should use golden tools only if you need fast gathering of resources. Gold is a soft metal, so it should only be used if there is nothing else and you need to mine something quick.
Don't try to kill mobs in the nether
Ghasts are tough to kill even with the help of a bow, and since zombie pigmen travel in groups, they are notoriously hard to kill as well. Do not venture into the nether without some armor and a bow.
Don't build TNT traps out of dirt
It is time consuming to rebuild dirt traps. Instead, try a more explosion-resistant material.
Don't use other tools for damaging mobs
Carry a sword with you until you craft a bow. Other tools may damage mobs, but they are inefficient, and will take more damage from the attacks.
Don't break snow with bare hands
This falls in with mining rare minerals, since you won't get snowballs without a shovel, this is pointless.
Don't re-power minecarts
When adding a new piece of coal to a minecart, it wastes the amount of power remaining in the cart. Minecarts will run for ~4 minutes on a straight track with one unit of coal, so remember to let the power run out before re-fueling.