Research problems in communication complexity
I'll start with answers to your general questions, then give one nice open problem with applications towards circuit complexity.

It's hard to say what areas a new communication complexity researcher should delve into, since easy problems have likely been solved already, and harder problems are hard ;)  One suggestion is to take known communication lower bounds (such as for Disjointness or Equality) and apply these to get new lower bounds in other areas of computer science.  New applications of communication complexity pop up all the time, and if you can find a niche no one's considered before, you might be able to leverage the existing bounds you're learning now rather than having to develop new lower bounds.  This will help you get some papers out while you're learning all about communication complexity ;)

Another suggestion is to work hard to ask new questions people haven't yet considered, or to think about new settings that are fresh and less well-studied. Multiparty communication might be a good area to look at---it is generally less understood than two-party communication, and (IMO at least) there's lots of fresh motivation for studying it (from e.g. multicore processors to wireless sensor networks)

For a concrete open problem, there's a nice one from multiparty communication in the number-on-the-forehead (NOF) model.  In the NOF communication model, there are $k$ players who wish to compute some function $f(x_1,\ldots, x_k)$ on $k$ inputs.   However, each player $PLR_i$ sees every input except $x_i$.  (You can imagine $PLR_i$ having $x_i$ written on his forehead.) Because lots of input is shared between players, it should be easier to compute functions of their input.  Conversely, proving lower bounds on the communication complexity in the NOF model is harder.

Here's a $\textbf{big open problem}$ in this area:  come up with some explicit function f(x_1,\ldots, x_k) for k=polylog(n) such that $\omega(polylog(n))$ communication is required in the NOF model.

Through a chain of results from the early nineties [Hastad-Goldman91, Beigel-Tarui94], it's known that such a result would imply that $f \not\in ACC_0$.  For this connection to circuit complexity, you're actually free to assume that each player sends a single message to some "referee" who compiles the messages to produce an output.  Lower bounds even in this restricted setting are far from what we'd like.  The current state-of-the-art lower bound techniques for the NOF model don't give lower bounds better than $\Omega(n/2^k)$ or thereabouts.  These become trivial once $k &gt; \log(n)$.