How do I totally remove a song from a playlist and add it to another playlist

Look in the Playlist, find the song and click on Edit/Delete or right-click/Delete.

You can add that song to as many Playlists as you want, in just the same way you added to the first Playlist. (Highlight the song, right-click/Add to Playlist.) You do not have to remove it from one Playlist simply to put it in another.

This is for Regular Playlists. However, you cannot delete a song from a Smart Playlist. You either have to change the attributes of the song so it is not included in the Smart Playlist, or change the rules of the Smart Playlist (which will probably affect other songs as well).