This is from my textbook:
I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, it would be $<P_0,P_1>=P_0(-2)P_1(-2)+P_0(-1)P_1(-1)+P_0(0)P_1(0)+P_0(1)P_1(1)+P_0(2)P_1(2)=0$, but isn't $P_0(t)=1$, $P_1(t)=t$ and $P_2(t)=t^2-2$? So it would be like to check sets in $\{1,t,t^2-2\}$ to see if they are orthogonal? that's very confusing.
You don't need to check whether $ \left\{ p_0,p_1,p_2 \right\}$ are orthogonal - you are already told they are. To find the best approximation you need to use the Gram-Schmidt process, which builds up this approximation by calculating the projection of $p$ on the subspace $\operatorname{Span} \left\{ p_0,p_1,p_2 \right\}$.
In order to calculate, say, $\left\langle p,p_2 \right\rangle$, simply use the formula you were given:
$$ \left\langle p,p_2 \right\rangle =p(-2)p_2(-2)+p(-1)p_2(-1)+p(0)p_2(0)+p(1)p_2(1)+p(2)p_2(2).$$
In order to calculate each factor of each summand, you need to simply substitute the corresponding point into the polynomial. For instance, $$p(-2)=5-\frac 12 (\color{#C00}{-2})^4,\;p_2(1)=(\color{#C00}{1})^2-2.$$