Hi Evan and welcome. You'll find that most around here use Photoshop. There are several free photo editor programs out there too . You might try
GIMP for Windows or
Paint.net
Photoshop and some other programs, use layers that allow you to move one image over the other. You will still need to eliminate the background of one image by erasing or extracting.
If your images are pixelated, you need to keep in mind the original file size and resolution of the pixelated image vs the other image you are placing it on. If you have a small Internet image and try to enlarge it to a large background you will loose the resolution. Hope this helps.