Select the iPod in iTunes. There is a row of buttons that starts with Summary. Click on Music next to Summary. This is the iPod's Music screen, where you tell iTunes how to sync songs (and playlists) to the iPod.
On the iPod's Music screen, check the box for Sync Music. If your entire music library fits on the shuffle, you can choose to sync Entire music library. Otherwise, you can choose to sync Selected playlists, artists, albums, and genres. Under Playlists, checkmark the ones you want on the iPod. Click Apply to sync your selection to the iPod.
Going forward, any changes to your iTune music library, that relate to your selection on the iPod's Music screen, sync automatically when you connect the iPod (or click the Sync button).