I've found that some good material on this subject can be found in Paul Halmos' A Hilbert Space Problem Book. Chapter 11, entitled Spectral Radius, contains some of the basic results in operator theory along with some specific problems about weighted shifts. In particular, the exercises work through the norm, spectral radius, point spectrum and the approximate point spectrum of a weighted shift. Some other exercises in this section utilize weighted shifts to construct examples and counterexamples to various questions presented.
One nice fact about weighted shifts is that given weights, $a_n,b_n$, if $|a_n|=|b_n|$ for all $n$ (either in $\mathbb{N}$ or $\mathbb{Z}$ depending on whether it is the unilateral or bilateral shift), then the weighted shifts determined by $a_n$ and $b_n$ are unitarily equivalent (This is exercise 89). A direct corollary to this is that the spectrum of a weighted shift is radially symmetric about zero.
The questions contained here don't answer in its entirety the question: "what is the spectrum of a weighted shift." However, they do contain some particularly illuminating facts and examples. A Hilbert Space Problem Book contains exercises, hints, and solutions. I highly recommend it for a better understanding of many aspects of operator theory.
Edit: After doing a little light reading, I came across an article that gives a much more complete picture of the the spectra of weighted shifts. The paper is entitled Approximate Point Spectrum of a Weighted Shift by William C. Ridge. It gives a very explicit breakdown of the parts of the spectra of weighted shifts based on different conditions on their weights. But, for the big picture, the spectrum of a unilateral weighted shift is always a closed disk centered at the origin; the spectrum of a bilateral weighted shift is always connected.