I'm a bit struggling to understand one part of how to actually transform the Christoffel symbols, in the case where the manifold is embedded in a higher dimensional space.
As an example consider the $\mathbb{R}^2$ as the standard manifold with the usual Euclidean metric and connection. Define an injective function $f:\mathbb{R}^2 \to \mathbb{R}^d$ which embeds the plane into a higher dimensional in an arbitrary non-linear way. According to the usual rules for change of variables of the Christoffel symbols we have that $$ \Gamma^k_{i j} = \frac{\partial f^k}{\partial x^m} \frac{\partial^2 x^m}{\partial f^i \partial f^j} + \frac{\partial f^k}{\partial x^m} \frac{\partial x^n}{\partial f^i} \frac{\partial x^p}{\partial f^j} \Gamma^m_{n p} $$ Since we are using the standard Euclidian connection in the coordinate system $x$ we have that $\Gamma$ vanishes everywhere so: $$ \Gamma^k_{i j} = \frac{\partial f^k}{\partial x^m} \frac{\partial^2 x^m}{\partial f^i \partial f^j} $$
My main question is how exactly is defined the expression $$ \frac{\partial^2 x^m}{\partial f^i \partial f^j} $$ in order to be able to calculate the Christoffel symbols?
You're not really doing a change of variables in this case since your map $\mathbb{R}^2 \to \mathbb{R}^d$ is not invertible (it won't be surjective), so you'll just have to compute Christoffel symbols the old fashioned way (if it were invertible then you would have a map $\mathbb{R}^d \to \mathbb{R}^d$ and the differential of this map would give you a $d$ by $d$ matrix. If you wanted to actually compute, e.g., $\frac{\partial x^m}{\partial f^i \partial f^j}$ you would have to write the differential at each point as a matrix (the Jacobian), invert this matrix, look at the $m,i$ entry and then differentiate that entry with respect to the $j$th variable).
Your map $\mathbb{R}^2 \to \mathbb{R}^d$ is giving you a parametrization for some surface in $\mathbb{R}^d$. We can then write the metric in these coordinates. We have $f(x, y) = (f_1(x, y), ..., f_d(x, y))$. In these coordinates our metric is a matrix of the form $g(x, y) = \begin{bmatrix} \sum_i (\frac{\partial f_i}{\partial x}) ^2 & \sum_i \frac{\partial f_i}{\partial x} \frac{\partial f_i}{\partial y} \\ \sum_i \frac{\partial f_i}{\partial x} \frac{\partial f_i}{\partial y} & \sum_i (\frac{\partial f_i}{\partial y}) ^2 \end{bmatrix}$ (I'm taking the image of $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ under $f$ and taking their dot products and recording it in a matrix). If the differential of $f$ has rank 2, then the matrix $g$ is everywhere invertible. Physicists would often write the matrix $g(x,y)$ as $g_{ij}$ and the inverse matrix as $g^{ij}$. Then the formula for Christoffel symbols using Einstein notation is $\Gamma_{k \ell}^i = \frac{1}{2} g^{im} (g_{mk,\ell} +g_{m \ell, k} - g_{k\ell,m})$ where the comma means partial derivative with respect to that variable.