If you want iTunes to sync automatically, you need to set iPod to Sync Music. The default setting after a Restore (in current iTunes) is to set iPod to use the manually manage method to load songs (drag and drop songs on iPod in iTunes).
Select iPod in iTunes (click its device button on horizontal bar) to show its settings screen in iTunes window. Along the left side (in the sidebar), under Settings, click Music. To the right, the iPod's Music settings screen is shown.
NOTE: For anyone else reading this, any songs on iPod that are not in your iTunes library (on computer) will be lost.
At the top, checkmark the box for Sync Music; this turns ON automatic syncing. Below that, if your music library can fit on your iPod completely, choose the option to sync Entire music library, and click Apply button. Otherwise, choose the option to sync Selected playlists, artists, albums, and genres. And select what you want from the lists below. A convenient way to set this up is to create a playlist in your iTunes library with all the songs you want on iPod. Back on the iPod's Music settings screen, under Playlists, find and select (checkmark) that new iPod Songs playlist. If you do it that way, you don't have to select anything else. Click Apply to sync your selection to iPod.
Going forward, iTunes keeps your iPod "in sync" with changes to your iTunes library, based on how you set up this screen. If you set it to sync a playlist, update that iPod Songs playlist in your iTunes library (add/remove songs). The iPod does not need to be connected. The next time you connect it (or click Sync if already connected), iTunes updates iPod with the same changes.