I have figured out that the density function for $(W(s), W(t))$ is: $$g_{s,t}(x_1,x_2)=\frac{1}{2 \pi \sqrt{s(t-s)}}e^{-\frac{x_1^2}{2s}-\frac{(x_2-x_1)^2}{2(t-s)}}$$ Also, the density function of $W(t)$ is: $$g_t(x_2)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x_2^2}{2t}}$$
My thoughts are that: $$g_{s|t} =\frac{g_{s,t}(x_1,x_2)}{g_t(x_2)}$$
Now the problem is, I do not know how to set up the integral for the mean,or if $g_{s|t}$ is a function of $(x_1,x_2)$ or just of $x_1$. I am getting back to Stochastics, after a long time. Greatly appreciate input on finding this mean.
Letting $B(t)=tW({1/t})$ for $t>0$ and $ B_0=0$, then it is standard result that $B(t)$ is also a standard Weiner process. Therefore, $$ E[W(s)|W(t)]=E[s B({1/s})|tB({1/t})]=s\cdot E[ B({1/s})|B({1/t})]\stackrel{\frac1s>\frac1t}=sB({1/t})=\frac{s}t W(t). $$