The minute I saw you were asking about dyeing wool, I had to sign in so I could respond!! Alas, as I read on, I realized you were looking for natural dyes, which I have no experience with.

I do know that people have luck using Kool-Aid as a dye (personally, I'd rather FD & C on my stuff, but I'm weird).
What I LOVE, though, are ACID DYES!!! Sound scary, but they are AWESOME!!! I buy mine from Dharma Trading, and they are so easy!! I don't know exactly what they are comprised of, but I prefer to do things more naturally, and apparently these dyes are pretty good along those lines (sorry--eloquency is eluding me . . . late night with the baby last night!).
Wish I could be more help about using beets and whatnot (which, IMO, would be the ONLY good use for beets!!

My boyfriend loves them, but I can't get over the dirt taste).