Before you can start playing singleplayer you need to create a new world. When you click on the Singleplayer button in the main menu it will take you to a screen where you can see all your current worlds, create new worlds and load existing worlds.
Click on "Create New World", you now need to choose a name for your world as well as the settings. There are currently two Game Modes: Survival and Creative. With Creative you have an unlimited amount of every block/item in the game. With Survival you need to harvest resources (Mine, Chop down trees, etc) and craft them into other items to survive against the enemy monsters that roam the land. The "More World Options" button gives you further settings like allowing cheats.
Once your happy with the settings you have picked click on the Create New World button. Minecraft will then build and load your world.
The contents in this section assumes you picked the Survival Game Mode, it doesn't apply to Creative. Its best to read the Crafting & Smelting page first if you have not already.
When you first enter your world you will have nothing in your inventory, to survive the night you will need to build a shelter. The most basic resource in the game is Wood, so it is important to find Trees. Once you have found a tree destroy its Logs and pick them up. Each Log can be crafted into 4 Planks making them a good building material. Craft the Logs into Planks and make yourself a small temporary shelter. You want to build your temporary shelter near where you spawned so that if you die it will be easy to find it again. You want to leave a hole 2 blocks high for the doorway. When it starts to get dark retreat to your shelter and place a single block in your doorway so there is a 1 block gap (Which allows you to see when its day again).
When you exit your shelter you need to be careful, not every enemy monster is killed by daylight... You now know the basics of surviving the night, what you do now is up to you!