I kind of understand what a connection on a vector bundle is. For instance, if we take the tangent bundle $TM$ to a smooth manifold $M$ then in coordinates I can think of it in terms of the Christoffel symbols $$ \nabla_{\partial_i} \partial_j = \Gamma_{ij}^k \partial_k. $$ I've just started thinking about fibre bundle connections and I'd like to be able to think about them in a similar way if possible. Obviously a vector bundle is a fibre bundle with linear fibres, so I'm wondering if I can think of a connection in terms of "Christoffel symbols" which now depend on where you are in the fibre. More specifically, if $N \rightarrow M$ is a fibre bundle and I have coordinates $(x,y)$ on $N$, where the $x$ coordinates correspond to $M$ and the $y$ coordinates to the fibre, can I say something like $$ \nabla_{\partial_i} y_j = \beta_{ij}^k(y_j) y_k, $$ where $\beta_{ij}^k$ is a function of $y_j$ (i.e. changes depending on where we are in the fiber)?
(Not sure if my notation is the best. Wanted to think in terms of how $y_j$ changes as we vary $x_i$.)
This does not quite work in the way you imagine. The problem is that the expression for the Cristoffel symbols you propose contains a summation (hidden in the Einstein sum convention) and this does not make sense if the fibers dont have a linear structure. Indeed, the interpretation of connections on fiber bundles in terms of covariant derivatives is more complicated. Given a (local) section $s$ of a fiber bundle $p:N\to M$, its tangent map $Ts$ maps $TM\to TN$ and $Tp\circ Ts$ is the identiy on $TM$. Hence for any tangent vector $\xi\in T_xM$, the value $T_xs\cdot\xi$ is a lift of $\xi$ to $T_{s(x)}N$. Using a connection, you can remove the horizontal part of this lift to get a "covariant derivative" $\nabla s\cdot\xi$, which now lies in the vertical subspace $V_{s(x)}N=\ker(T_{s(x)}p)\subset T_{s(x)}N$. So the covariant derivative of $s$ is a vector bundle map from $TM\to M$ to $VN\to N$ with base map $s$.
I don't think that a representation in terms of Christoffel symbols makes much sense in this setting, since the linear structure you bring into the game via the fiber coordinates is completly arbitrary. This means that there is no natural way to recover $\nabla s$ from the local coordinatate functions $y_i$ describing $s$ (since $\nabla$ is not compatible with the linear structure coming from the chart).