  We present a complete and detailed derivation of the finite lattice spacing
corrections to staggered fermion matrix elements. Expanding upon arguments of
Sharpe, we explicitly implement the Symanzik improvement program demonstrating
the absence of order $a$ terms in the Symanzik improved action. We propose a
general program to improve fermion operators to remove $O(a)$ corrections from
their matrix elements, and demonstrate this program for the examples of matrix
elements of fermion bilinears and $B_K$. We find the former does have $O(a)$
corrections while the latter does not.
