>The carrots already pre-packaged and shredded just don't taste
>as good, they taste kind of hard, it makes me think they have
>lost some of their nutrients.
They definitely have. Anytime food is cut (like shredded carrots, those pre-sliced apples, etc.) and exposed to air, more nutrients are lost. Shredded carrots would be particularly bad, since much more of their surface area is exposed.
A food processor is one of the most useful kitchen tools I own--it makes hummus dip and guacamole and bean dip and such in a fraction of the time it would take to prepare without it.
If you don't want to invest in a food processor, there are some hand-crank versions (like Salad Shooter) that are faster than grating, but much less expensive than a food processor.