Q: ipod shuffle

If your iPod shuffle is the current 4th gen (or 3rd gen) model, to play songs in a specific order, you need to put them on a playlist in iTunes (in the desired order).  You then sync that playlist to the shuffle by selecting it on the shuffle's Music settings screen in iTunes.  This is the same way you use with the "big" iPods (that have a screen), to play a song list in a specific order.

On an iPod with a screen, you select the playlist to play on the screen.  On a shuffle, you use the VoiceOver feature to select the playlist.  This document describes how to set up and use VoiceOver on a shuffle

iPod shuffle (4th generation): Enabling and using VoiceOver - Apple Support

To play selected playlist in playlist order (as shown in iTunes), put the shuffle's power switch in the play-in-order (middle) position.

Once you set up automatic syncing for the playlist, you can update the playlist (add/remove/re-order songs) in your iTunes library.  The shuffle does not need to be connected.  The next time you connect the shuffle (or click Sync if already connected), iTunes updates the shuffle with the same changes.  You can set up automatic syncing for multiple playlists, and select the one you want to play using VoiceOver on the shuffle.

Please post back with questions about setting up automatic syncing on the iPod's Music settings screen in iTunes (or other questions about the above).

NOTE:  If your shuffle is an older model (1st or 2nd gen), it works differently.