You can eat them raw. I just had a half one on my salad yesterday and the other half today. I didn't even peel it. Just scrubbed it well and went to town!
Beets and carrots is a lovely combination, IMO. I just discovered it and I'm hooked.
Beets do tend weep juice everywhere and that juice *does* tend to stain, so be careful with not only grating your beet, but also what you wear while you're grating that beet (I think I ruined a shirt last night... ).
If you got the tops, too, lightly wilt that with a bit of garlic or chop it up and add it to your salad.