For a project it has been specified that my images need to be at least 6MB. However, whenever I copy or move them around they dramatically decrease in size - and that is even before I have edited them which compresses them further.
You've been given a poorly considered requirement. Most likely, the intent was to specify some sort of minimum resolution for your images, but some images compress more readily than others. Double check that the requirement really is for 6 megabyte images and not 6 megapixel images — it'd make much more sense to specify the file size in terms of pixels, since the pixel dimensions will remain the same regardless of compression. If they really did say megabytes, you might ask for some clarification about what they really mean, or what image dimensions they'd like.
I have tried shooting in JPEG and RAW+JPEG but they still go very small when I want to move them off my camera memory card.
It sounds like you might not actually be importing the RAW version of the file, or you might be importing it but only looking at the JPEG image. JPEG files are generally a lot smaller than their RAW counterparts.