Check equivalence of quadratic forms over finite fields How to check whether the two quadratic forms \begin{equation} x_1^2 + x_2^2 \quad \text{(I)}\end{equation} and 
\begin{equation} 2x_1x_2 \quad \text{(II)} \end{equation} 
are equivalent on each of the spaces $\mathbb{F}_3^2\, \text{and}\,\mathbb{F}_5^2$? 
I know that these forms correspond to the two matrices \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} in case (I) and 
\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} in case (II). But what do I do with those and what difference do the different fields make? 

You've wrtten the other way round. For the quadratic form $ x_1^2 + x_2^2 $, the corresponding matrix is the identity while that for $ 2x_1x_2 $ is $ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} $. The forms are equivalent iff the matrices are congruent. (over $ \mathbb{F}_3 $ and $ \mathbb{F}_5 $) 
So, suppose $ A = P^TIP = P^TP $ over $ \mathbb{F}_3 $ for some invertible $ P $. Then, taking determinants gives $ -1 = ( \det P)^2 $, which is a contradiction as $ -1 $ is not a quadratic residue modulo $ 3 $.
Over $ \mathbb{F}_5 $, you can check that they indeed are equivalent by the matrix $ P = \begin{pmatrix} 1 & 3 \\ 3 & 1 \end{pmatrix} $ as we have, $ (3x_1+x_2)^2 + (x_1+3x_2)^2 = 2x_1x_2 $ in $ \mathbb{F}_5 $.
