If you right-click on the image that you've found that you wish to work with, you should get a listing (referred to as the Context Menu). One of these is to save the image (Save Picture As):
The recommended place to save your downloaded images is in the directory where all your other pictures are at, but if it makes it easier for you to save to your desktop then that will be okay as well.
In Paint.net go to the menus at the top of your screen and go to "Layers". You should see an option to "Import from File". Select that and then navigate to your downloaded image and select it. Once the image is on your new layer, in the Layers Pane on the right side of the screen using the down arrow at the bottom of it, click on it to move your layer down (thus putting the layer "behind" your background). If your Layers Pane isn't currently showing, press "F7" to open it.
EDIT: See the right side of the screen shown below:
To take a screenshot of your current project, press the "Prt Sc" (print screen) key on your keyboard. Next go to the menus and select File>New (or you can use the keys "Ctrl + N"). Once a new canvas has opened up, press "Ctrl + V" to paste your screenshot to the new canvas.