Why is the blow up of a reduced scheme reduced? This is in Vakil's notes (22.2.C) right after he gives the universal property of the blow up involving Cartier divisors, but before the explicit construction of the blow up using Proj. Using only the universal property would be preferred, but an approach that uses the construction of the blow up involving the Rees algebra is better than nothing.
I tried the second approach, but I didn't make it clean enough for me to believe its correctness or remember the idea for future reference.
When I googled the question just now, I got http://stacks.math.columbia.edu/tag/0808, and if you follow the links, you end up here: http://stacks.math.columbia.edu/tag/052S, where they omitted the proof.
Recall that any localization of a reduced ring is reduced. Since the Proj of a graded ring $R$ is covered by open affines which are Spec's of degree zero parts of various localiations of $R$, if $R$ is reduced then so is its Proj.
In the case of blowing up Spec $A$ along an ideal $I$, we have $R = A \oplus I \oplus I^2 \oplus \cdots,$ and you can easily check that if $A$ is reduced then $R$ is.
Since your question is local on the scheme being blown-up (i.e. the preimage of an affine open Spec $A$ in the scheme being blown up is the Proj of $R$ above, for an appropriate choice of $I$), we see that any blow-up of a reduced scheme is reduced.