As Fernando Muro points out in the comments, Sylvester's law of inertia is probably the easiest way to determine the signature. You diagonalize the symmetric matrix by the Gram-Schmidt process. This is essentially as easy as Gaussian elimination.
A symmetric matrix is equivalent to a bilinear form, with basis $\{v_1, \ldots, v_n\}$, and matrix entries inner products $\langle v_i,v_j\rangle, i,j\leq n$. If $\langle v_1,v_1\rangle \neq 0$, then replace $v_i$ with $v_i-v_1 \langle v_1,v_i\rangle/\langle v_1,v_1\rangle$, and proceed by induction (this has the effect of performing the same row and column operations, so as to keep the matrix symmetric). The only variation on the standard Gram-Schmidt is what to do when $\langle v_1,v_1\rangle=0$, which clearly means the form is indefinite. If $\langle v_1, v_i\rangle=0$ for all $i\leq n$, then you can proceed by induction. Otherwise, by permuting indices, you may assume that $\langle v_1,v_2\rangle \neq 0$. Replace $v_1$ with $v_2+tv_1$, such that $\langle v_2,v_2\rangle+2t\langle v_1,v_2\rangle\neq 0$, and proceed as with the usual Gram-Schmidt. Once you have diagonalized in this way, Sylvester's law implies that the signature of the diagonal matrix is the same as that of the original one.