You should certainly read "Two Dogmas of Empiricism" -- it's one of the best-known papers in analytic philosophy and can be said to have set a large part of the agenda for Anglo-American philosophy since its publication in 1951. Better to read the paper itself, anyway, than to read things things about it, as you evidently have.
Circularity doesn't itself play that much of a role in Quine's paper. His attack on the analytic-synthetic distinction is waged on two broad fronts (to oversimplify a bit), a logical one and a linguistic one. The logical one is largely implicit in "Two Dogmas" but gets more attention in some later writings of Quine's. It derives from Gödel's first incompleteness theorem, which showed that there can be (under certain conditions) true sentences in an axiomatically defined language not provable from its axioms (i.e. not "analytic" as that is understood by e.g. Frege). So if you want a distinction between sentences that are simply an artifact of the language you've chosen and those that actually convey some empirical information about the world, you'll need a criterion other than provability. That's hard, and no one has come up with anything nice and simple that applies across the board and not just in special cases. For Quine, that was reason enough to give up on the distinction altogether. Others (including scientists such as Einstein) thought the distinction was absolutely critical to science and weren't too concerned that it couldn't be pinned down by a precise criterion.
Quine's other attack was on a quasi-empirical front; in ordinary language, he claimed, you can't find any such analytic-synthetic distinction, you can't find a built-in "criterion of analyticity." (This also gets more attention in his book Word and Object a few years later.) It's a distinction that has to be imported into ordinary language from our constructed logical and mathematical languages. However Rudolf Carnap, against whose ideas "Two Dogmas" was mainly directed, had no problem with that, and this part of Quine's critique no longer gets so much attention.
Your final question about circular definitions is hard to answer briefly. There are some philosophers who think that the point of philosophical analysis is not "reductive" analysis, whereby you analyse (or define) all concepts in terms of more basic ones and so get down to some foundation or small set of basic concepts (as in Bertrand Russell, as he's usually seen), but rather "connective" analysis, whereby we figure out how the various concepts we use in our language fit together. For those philosophers (Peter Strawson, for instance, who coined the term "connective analysis"), circularity in some wide sense is not a bad thing.
In most of science and mathematics, on the other hand, circularity is obviously a defect because a circular definition does no work. To say that table salt is composed of two elements, sodium and chlorine (each of which we know a lot about), in a certain electrochemical combination of units or atoms (one each), is informative and can predict certain things about table salt that you couldn't otherwise, whereas to say that table salt is table salt, or is salty by virtue of having a tendency to saltiness (extreme version of a circular definition) gets you nowhere and predicts nothing.