Exercise:
What is the expected number of dice rolls until the sum of the rolls is a multiple of $7$ ?
Attempt:
Let $S_n$ be the sum of the results of $n$-roll dices and $T_k = \inf\left\{ k >0 : S_k \; \text{multiple of 7}\right\}$.
Let's assume the Markov Chain $S'_n = S_n\mod 7$ and then it will be $T'_k = \inf\left\{ k \geq 1 : S'_k = 0\right\}$.
Since we are interested in the expected value $\mathbb E \left[T'_k\right | S'_k = x]$, I know that we have to work with $g(x) = \mathbb E \left[T'_k\right | S'_k = x]$ where $g(x)$ is a solution to the problem $Lg(x) = -1$ and then find $g(0)$, but I am having a hardtime constructing the state space and understanding the modeling of the problem.
I would really appreciate any thogough explanation as Stochastic processes isn't really my field.
Your definition $T_k=\inf\left\{k\ge0:S_n\;\text{multiple of 7}\right\}$ doesn't make any sense. $k$ is a free variable on the left and a bound variable on the right; $n$ and $k$ are not related; and if you allow $k=0$ the infimum is trivially $0$. So I’ll make three changes to it and assume that you meant $T=\inf\left\{k\gt0:S_k\;\text{multiple of 7}\right\}$.
If so, you don’t need a Markov chain for this problem. On every roll except the first roll, if the experiment has not ended yet, you have probability $\frac16$ of reaching a sum $\equiv0\bmod7$. Thus the expected number of rolls beyond the first is geometrically distributed with parameter $\frac16$, and the expected total number of rolls is $1+\frac1{\frac16}=7$.