Create superflat map in Minecraft with trees, grass, etc

You can add structures such as trees and grass (and others) using the Customize interface. As of 12W40A, you have to use More World Options.... From there, set the World Type to Superflat, then click Customize and then Presets. You'll be presented with a box that lets you share presets. The format for these presets can be found on the wiki.

To create a superflat world that also generates trees, flowers, and tall grass, use the following preset:

2;7,2x3,2;3;decoration

This will generate a world with a layer of bedrock, two layers of dirt above that, and then one layer of grass above that. Trees, flowers, tall grass, and so on will be generated. The trees will be fairly spaced apart.

Note that villages cannot be generated here, since they only generate in plains or desert biomes, and the Extreme Hills biome used to add the trees, flowers, and tall grass does not allow villages to spawn.

However, structures such as Abandoned Mineshafts and Strongholds can still be generated. Simply add ,mineshaft, ,stronghold, ,dungeon, etc. at the end of the preset. There are various parameters that can be added, such as ,mineshaft(chance=0.1). These parameters are detailed in the wiki.

If you want more trees, you can change the 3 to a 4 to use the Forest biome. This will generate trees far more densely.