Markov Chain vs Bayes Net
First of all, I'd disagree that Markov Chains are dealing with a &quot;single type of variable&quot;. If you look at the formal definition of a Markov Chain, you'll see that variables $X$ are random variables. And random variables are defined over arbitrary (well, measurable) sets of possible outcomes. So your $X$ can not only be from $\{sun,rain\}$ set, but if could be from a Cartesian product of $\{sun,rain\}$ and $\{windy,cloudy,calm\}$ and temperature from $[-60,60]$ interval.
About the relation between Markov Chains and Bayes Nets, I'd say that there is a common framework that lets you understand relationship between those (and, in fact, many other probabilistic structures). In all cases we have a collection of random variables - for Markov Chains these are $X_0,X_1,X_2\dots$ And for Bayes Nets lets call them $A,B,C,D,E,F$.
In both cases we are interested in the probability distributions over the whole collection of variables, called joint probability distribution:
$$P(X_0,X_1,X_2\dots)\qquad\text{and} \qquad P(A,B,C,D,E,F)$$
The problem is that trying to represent this distribution (say on a computer) is impossible for any reasonably complex problem - the number of possible combinations of variables grows exponentially with the number of variables. So we are coming up with a way to factorize these joints into smaller, manageable multipliers.
For the Markov Chains, the factorization property is
$$P(X_0,X_1,X_2\dots) = P(X_1|X_0)P(X_2|X_1)P(X_3|X_2)\cdots$$
For the Bayes Nets, the factorization property could look like this (for a particular dependency graph that I just made up):
$$P(A,B,C,D,E,F) = P_a(A)P_b(B)P_c(C|A)P_d(D|A,B)P_e(E|C,D)P_f(F|E)$$
In my opinion, considering the joint distribution and then seeing what factorization structure is imposed on it by your framework is a good place to start studying it.