How to compute a connection in practice?

Question

I just started learning about Riemannian geometry. I came across the notion of a connection $\nabla$. First, does the connection depend on the metric? If, yes, how does one compute the connection from the metric? For example, if I consider the half-plane $\{(x,y)\in \mathbb{R}^2 : x>0\}$ with the metric $\langle u,v \rangle_{(x,y)}=u^T\begin{bmatrix}1/x^2& 0\\ 0 & 1/x^2\end{bmatrix} v$. Then what would be the connection $\nabla$ associated to this metric (I mean how do I compute it in this case and in general)?