Good introduction to "forcing" and "inner models"? I've occasionally come across the use of forcing, e.g., JDH's exploration of the modal logical of forcing. I know it is a massively important proof technique for, e.g., independence proofs. I've also come across so-called "inner models" (do you "force" inner models?) like countable models of set theory.
All of this stuff seems pretty cool, but right now I just have a relatively small background in math logic and set theory (a couple of courses and some independent work; problem is that my knowledge is more wide than deep).
So, what is a (relatively) accessible entry point into the literature on forcing and inner models?
(Though this is tagged a reference request, I would certainly love to have users' own input beyond supplying a reference as well. Particularly important insights, etc.)  
 A: I can only give you a sketch since I am not an actual expert in either topic.


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*There is no royal road to these topics. These are two separate subjects in set theory. Both are needed to fully establish independence results and relative consistency results. Each one can take many many years to learn and understand properly.

*If $V$ is the universe of set theory (i.e., we work internally to a model of set theory), an inner model is a transitive class containing all the ordinals which is a model of $\sf ZF$ or $\sf ZFC$ (depending on the context). As far as $V$ is concerned an inner model is never a set.
For example, $L$ is always an inner model of $V$. The technique was used to establish the consistency of the axiom of regularity by von Neumann; and by Godel to establish the consistency of the axiom of choice and the continuum hypothesis. Since then this method has been developed by many people and we use it for similar relative consistency results.
One example is the following. Suppose that you are in a model of $\sf ZFC$ such that every $\Sigma^1_3$ set of reals is Lebesgue measurable. By a result of Shelah, we can construct an inner model that has an inaccessible cardinal. Thus we establish that if "sufficiently complex" sets are Lebesgue measurable, then we had to assume the consistency of inaccessible cardinals.
A recommended roadmap to inner models is an old questions by yours truly about the subject, with a wonderful answer by Andres Caicedo. A newer source to consider as well is Schindler's new book about set theory. Although I cannot really tell you what and how to read it. Admittedly, I never followed the entire roadmap.

*Where inner models dig "into the universe", forcing extends the universe by adding new sets in a careful way, and examining the consequences of these new sets on the universe. There are many approaches to forcing, some include the use of transitive models, others do not. It is useful and important to learn the various methods, but it can take quite some time.
As mentioned by Stefan, Good resources for studying independence proofs contains some information about studying forcing (and using inner models to establish choice-related consistency results). The general outline is that Halbeisen, Kunen and Jech are the general resources for studying forcing. They take on slightly different approaches, and you should at some point or another visit all three (perhaps with less attention to details once you're sufficiently trained).
All in all, forcing and inner model theory make more or less all modern set theory. So there are many things to learn about them. But starting with the basics, understanding $L,\mathrm{HOD}$ and the relative constructibility inner models; as well basic forcing facts and applications are also necessary. After that, you might want to regroup and decide which topic is next (large cardinals, for example).
