I don't see the add button in apple music

Make sure ‘Show Apple Music Features’ is activated in iTunes Preferences.

Search any song or album in Apple Music (not the Store tab!)

For songs, right-click and select Add to Library or Add to Playlist from the contextual menu.

For Albums, from the collection of matches page, right-click an album and select Add to Library or Add to Playlist from the contextual menu. Or ‘open’ an album and use the More icon to get the same menu choices.