Downloading an app from the AppStore is very easy to do. You can either download them directly on your iPod touch (as long as you're within wi-fi), or download them through iTunes. Many of the apps are free; some do cost money (some as low as 99 cents).
After downloading it, you don't need a wi-fi connection again to use it, unless the application requires it. For example, to play most games on the iPod touch, it doesn't need wi-fi. On the other hand, if you download the Facebook app, it does.