Find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$. 
If $X_1,X_2\sim \text{Normal} (0,1)$, then find $Y=f(X)$ such that $Y \sim \text{Uniform}(-1,1)$.

I solve problems where transformation is given and I need to find the distribution. But here I need to find the transformation. I have  no idea how to proceed. Please help.

Why not use the probability integral transform?
Note that if 
$$
  F(x)
= \int_{-\infty}^x \frac{1}{\sqrt{2\pi}} e^{-u^2/2} du
$$
then $F(X_i) \sim U(0,1)$. So you could take $f(x) = 2*F(x) - 1$.
