Rescaling – How to soften a pixelated image before printing?

pixelated image that needs to be softened

I have a lot of images like this one that are completely pixelated, and my friend wants to make large prints to hang in her house. How can I soften them so they don't appear pixelated? If I apply anti aliasing, would that be enough? If so, what kind of anti aliasing? Is there a good way to convert them to vector graphics? What level of image quality do people usually need to make a large print look good?

To be clear, these images have several resolutions, including 800×600 and 960×720. Each pixel acquires one of exactly three colors. It has not yet softened or averaged in these images.