I was wondering if it was possible if I had an image like: 
I would be able to change certain parts of the images colors. For example if I wanted the bow green and the present red with yellow stripes, would I have to make a new image that had that or is there a way to program something (elegantly) along the lines of that? I’m just asking to see if its possible and if it is, what language would be best to do this?
Keep in mind this would be a feature on a website.
That’s not going to be easy the way the image is. You have no way to tell the computer which part is the bow, which part is the stripes, and which part is the box. However, if you pre-colored them, you could do a color replace using GD library or imagemagick pretty easily. You’d do this in PHP. Here are some examples of how you could do it, I’d personally go the imagemagick route.
How can I replace one color with another in a png 24 alpha transparent image with GD
http://www.imagemagick.org/Usage/color_basics/#replace
(this example even has a similar gift box as the usage case, hehe)