There is the following result of Wolke from $1967$ (which is perhaps not the best, but quite good). Theorem: Let $p$ be an odd prime, and $L(s,\chi)$ the $L$-series for the Dirichlet character $(n/p)$. If $t(p)$ is a positive function with $L(1,\chi)>t(p)/\log(p)$, then there are absolute constants $c_1,c_2>0$ with $$ r(p)\le c_1 p^{c_2/\sqrt{t(p)}} $$ Here one should mention the result of Elliott: for an integer $k\ge 0$ and real $c>0$ we have $$ L(1,\chi)\ge \frac{c(\log \log p)^k}{\log p}. $$ For $n(p)$ see the the report of Terence Tao, and http://www.math.ubc.ca/~gerg/teaching/613-Winter2011/LeastQuadraticNonResidue.pdf.

There is the following result of Wolke from $1967$ (which is perhaps not the best, but quite good). Theorem: Let $p$ be an odd prime, and $L(s,\chi)$ the $L$-series for the Dirichlet character $(n/p)$. If $t(p)$ is a positive function with $L(1,\chi)>t(p)/\log(p)$, then there are absolute constants $c_1,c_2>0$ with $$ r(p)\le c_1 p^{c_2/\sqrt{t(p)}}. $$ Here one should mention the result of Elliott: if we have for an integer $k\ge 0$ and real $c>0$ $$ L(1,\chi)\ge \frac{c(\log \log p)^k}{\log p}, $$ then for every $\epsilon >0$ we have $r(p)\le c(\epsilon)p^{1/4(1+\epsilon)(k+2)^{-1}}$.

For $n(p)$ see the the report of Terence Tao, and http://www.math.ubc.ca/~gerg/teaching/613-Winter2011/LeastQuadraticNonResidue.pdf.