When you go into the Playlists menu of your iPod Classic, you should see a list of the Playlists you have downloaded from your iTunes Library - and the On-The-Go Playlist. If you can only see the On-The-Go Playlist, that suggests that you have not successfully downloaded any Playlists.
Playlists are separate from the songs. The songs, albums etc. etc. will be listed under those menu headings on the Main Menu.
The On-The-Go Playlist is a feature that allows you to create Playlists on the iPod. Initially, and after each Sync with iTunes, the On-The-Go should be empty. If you create one, you can save it on the iPod and it will be transferred back to the iPod at Sync time and renamed as On-The-Go X (X is the next available number). If you create and save one on your iPod, it will be named on the iPod as New Playlist X, but renamed as On-The-Go X when Synced with iTunes.