When I selected a part or a clip from a recording and give that part a label, the label does not move sync with the clip. Is there a trick to do that?
Select in the label track as well as the audio track, or enable Tracks > Sync-Lock Tracks.
But how can I store those syncs and get them back when I reopen Audacity?
You may need to describe exactly what you are trying to do. Labels are saved in the Audacity project. Labels are not included in exported files.