If you started a new single player game and didn't change any of the difficulty settings, the first thing you need to do after the world is created is figure out how to survive your first night: at night is when all the really bad stuff that'll kill you comes out.
The trick to doing that involves securing two things: light and shelter. Bad stuff doesn't spawn near light, and shelter will keep that which has already spawned from coming to kill you.
To build a shelter and a light source, you'll need some basic tools and materials. Look around you to see if you can see some trees:
If you left click and hold on the trunk of the tree, you'll be able to cut it down and get a log:
Keep cutting down trunks of trees until you have about a half-dozen logs. Once you have that, open up your inventory (press the "I" key):
Here, you should see the wood you just collected. Click on the wood and move it to the crafting slots at the top of the inventory screen. The slot to the right of the arrow will display a new crafted item, the wood plank:
If you click on the wooden plank, you can move it into your inventory, and the number of logs you have will decrease. Congratulations, you created your first item! Keep creating wooden planks until you run out of logs.
Now, take your wooden planks and put them into the crafting area of your inventory so there's a wooden plank in each slot (you can split stacks of items by right clicking on them). This will let you make a crafting table:
The crafting table will let you create things with 9 slots instead of the 4 your inventory allows. Move the crafting table to one of the bottom slots in the inventory (your action bar) and close the inventory window. If you then select the slot with the crafting table (if you put it in the second slot, you can select it with the "2" key), you can right click anywhere to place the table.
Once you place the table, right click on it to open the crafting screen:
Now, take a few of your wooden planks and lay them out in the grid so one is above the other so you can create wooden sticks:
Two planks equals four sticks. Create 12 sticks (so, use six wooden planks).
Now, you'll want to create a pickaxe, which will let you mine stone and coal. Move two sticks and three wooden planks into the crafting table and arrange them thusly:
Equip your pickaxe into one of your open action bar slots. Now that you can mine stone, you'll need to find some coal so you can make some torches. Look around you to see if you can see some mountains, particularly ones with bare stone and ideally ones with little black dots on them:
Depending on your luck when you spawned, you might be really close to some coal or you might have to walk around a while to find some.
Before you head off, make sure you collect your crafting table so you don't have to make another one later. To do that, left click and hold on the crafting table until it gets "destroyed" and walk over it to collect it.
If you don't find any coal after a few minutes of searching, try to find a secluded place with some bare stone that you can start mining into. Coal can appear almost anywhere underground, so you might get lucky and find some after digging into the side of a mountain for a bit.
Mining works just like cutting trees down: left click on the stone until the block is destroyed and you get a piece of cobblestone (or, if you mined coal, a piece of coal). Just make sure you've equipped and are using your pickaxe or it'll be an exercise in frustration.
If, after a few more minutes of mining, you still haven't found some coal, you can use the new-fangled 1.2 method of getting coal using a furnace. First, go get a couple more logs from nearby trees. Then, place your crafting table down (you did remember to collect it like I said to above, right?) and open it up to create a furnace with the cobblestone you no doubt collected via mining:
The furnace is a similar item to your crafting table: place it down and right click on it to open it. There, you'll see three slots. The bottom slot is the fuel source, the top slot is what you want to "smelt", and the right slot is what is created after smelting. In this case, you'll want to use a log as both the fuel source and the object to be smelted, which will create charcoal, a poor man's version of the coal you can mine:
Once you have coal (either by finding it or by making it), make some torches with the coal you found and the sticks you have:
Now, start digging inward until you've carved off a nice little nook or cave. Make sure you keep the opening small: you don't want bad things coming in. Equip your torches and right click on the sides of the walls to place them. By this point, either the sun has already gone down, or it's late in the day.
Close off the opening of your cave so nothing can come in and wait for daylight. To do that, find some of the cobblestone and/or soil you have from digging in your inventory and equip it like you did with the torches and pickaxe. Right-clicking with a block of stuff will place said block on the ground.
Congratulations, you survived your first day and got to play around with the basics of Minecraft. Now, you can start to explore the world around you and figure out what else you can make. Minepedia, where most of the above images came from, contains a wealth of knowledge about the game and what you can do.