Every time I print a screen and paste it into Word, the image is not only resized to the page but the quality is kept. When I paste it into Illustrator or InDesign, the quality is butchered and I have to resize it to the page butchering it even more. Is there anyway I can remedy this or will I have to do my work on Word..?
Don't copy/paste screenshots immediately in InDesign (and perhaps the same goes for Illustrator). Save your image as a PNG (if you don't mind RGB images) or as a TIFF (if you want to have it professionally printed, so it can be converted to CMYK if required). Then use the Place command in InDesign to "import" it.
If you find the images still look 'butchered', don't worry about that; InDesign uses by default a low-resolution proxy image to speed up drawing. You will find the original image quality restored when outputting to a PDF.
The reasoning not to include images in their entirety into InDesign is the following: