Equation of tangent plane for a differentiable $f:\mathbb{R}^2\longrightarrow \mathbb{R}^2$

For any differentiable function $f\colon\Bbb R^2\to\Bbb R^m$ you will have precise the same formula. Now every term on the right-hand side is a vector in $\Bbb R^m$ and you add the three vectors. The tangent plane is a $2$-dimensional plane in $\Bbb R^{m+2}$.

EDIT: In particular, if $f(x,y) = \big(f_1(x,y),f_2(x,y),\dots, f_m(x,y)\big)$, then you just differentiate component by component. So your tangent plane will be given by \begin{align*} z_1 &= \frac{\partial f_1}{\partial x}(x_0,y_0)(x-x_0)+\frac{\partial f_1}{\partial y}(x_0,y_0)(y-y_0) \\ z_2 &= \frac{\partial f_2}{\partial x}(x_0,y_0)(x-x_0)+\frac{\partial f_2}{\partial y}(x_0,y_0)(y-y_0) \\ &\vdots \\ z_m &= \frac{\partial f_m}{\partial x}(x_0,y_0)(x-x_0)+\frac{\partial f_m}{\partial y}(x_0,y_0)(y-y_0). \end{align*}