Let's say $|a\rangle$ is the pure state qubit and $|b\rangle$ is the bell state.

Putting $|a\rangle$ through the Pauli channel depends on which of the Pauli gates (which of the $\sigma_j$) you want acting on it. Once you choose this (call it $\sigma_a$), apply $2\times2$ Identity operators to it on either side until it's the same size as $\rho$, then just do the matrix multiplication: $q_a\sigma_a \rho \sigma_a$, where the $\sigma_a$ are both implicitly including enough identity matrices to make them the same size as $\rho$.

The Bell state will involve two qubits, so you can choose up to *two* Pauli operators ($\sigma_b$ and $\sigma_c$) that will act on it. You take the result of the previous matrix multiplication and do another matrix multiplication.

I can fill in more details, but it is starting to feel like I'm doing homework for you so let's see if you can follow what I've written and if you have follow-up questions in the comments I can modify my answer.

Thank you so much for your help! Actually I am working on some protocol and was interested in learning the fundamentals rather than just using Mathematica to solve even such small things. – Shivang Srivastava – 2019-10-06T11:29:52.560