There is no 'trace tool' in paint.net.
So you either do it this way viewtopic.php?f=15&t=4841 and then just put it on your new background.
Or you use the eraser tool and erase everything you don't need in the picture and then just add your new background (import from file, then move that layer down under the other one). I hope this makes sense.
You could use the lasso but it's easy to mess up that way.