App Store for iPod Touch 1st
To use the App Store on your iPod 1st-gen, you will have to purchase the iOS 3.1 update for iPod Touch (1st Gen), which is $4.99. 

This link will take you to the App Store on iTunes which will allow you to purchase and download the software update. The App Store was released in iOS 2.0.1 for iPhone 3G, so this update is required.