Start building your library by adding your favourites to My Music! Next to any track, tap the (•••) menu to the right of the song name > Save to Library. (On iOS, you can also set up a Quick Action to add things to your library in one tap. Do this from Settings > Quick Actions > Like!)
To delete, swipe left. (Reminder - when you delete an Album, Artist, or Playlist, all the corresponding songs will be removed from your Songs folder.)