rexrowland, as you said enlarging a small logo can give terrible results. It depends on the original source and how much you enlarge it. Vector based images (EPS, SVG, AI) can be enlarged without problems.You would have to try it and determine if the results meet your needs. There is information on the web or on YouTube about converting to vector from raster images. I don't know how well they work. There are web based programs that somebody on this forum used but I don't remember the site. Inkscape is a free vector graphics program. GIMP and Paint.Net can also be used and are free. Paintshop Pro, Adobe Photoshop and Adobe Illustrator can also be used. If you are referring to Windows Paint then there are many, many better alternatives. You can start a new post and ask more questions or for advice since this is not on topic for this post. Here's a simple raster vs vector video.
https://www.youtube.com/watch?v=MoCSwF0n-io