From http://en.wikipedia.org/wiki/Firefly_%28TV_series%29 - Synopsis

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The series takes place in the year 2517, on several planets and moons. The TV series does not reveal whether these celestial bodies are within one star system, and does not explain whether Serenity's mode of propulsion is faster-than-light, only that it is a "gravity-drive". The film Serenity makes clear that all the planets and moons are in one large system, and production documents related to the film indicate that there is no faster-than-light travel in this universe. The characters occasionally refer to "Earth-that-was" and in the film, it is established that long before the events in the series a large population had emigrated from Earth to a new star system in multi-generational spaceships : "Earth-that-was could no longer sustain our numbers, we were so many." The emigrants established themselves in this new star system, with "dozens of planets and hundreds of moons." Many of these were terraformed, a process in which a planet or moon is altered to resemble the Earth. The terraforming process was only the first step in making a planet habitable, however, and the outlying settlements often did not receive any further support in the construction of their civilizations.

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The “one large” system is actually the one containing the “Core” planets that founded the Alliance. If all the planets and moons were in the same system, even a big one, surely the outlying worlds would be a lot colder and more forbidding. The “they don’t go faster than light” hypothesis only really works if Firefly and Serenity is set a lot nearer to the Galactic Core, except it would take mankind thousands of years to do so: what’s the point? Also, nearer the Galactic Core it would never really be dark as the stars are so close together.

Navigation and communication in ‘Firefly’ seem to involve something referred to as the “Cortex”, which everyone, Alliance and Reavers included, seems to use. The use of a faster-than-light communications grid would be essential if traveling faster than the speed of light, otherwise you’d only know where you’ve been, not where you’re going. In order to use the Cortex, every star ship constantly broadcasts its own beacon. Finding a major planet or port is then quite straightforward: look for lots of beacons!

In terms of new colonies being established by the Alliance, a probe is probably sent at sub-light speeds to a target system with a terraforming crew close behind.

It is conceivable that Mankind was limited to traveling at sub-light speeds until after the Core planets were terraformed, unless the hundreds of off-world colonies that would naturally be established on habitable planets became the “Independents” in the War of Unification.

However, saying all that it is more than equally apparent, despite some inconsistencies, that the events in ‘Firefly’ and ‘Serenity’ are, after all set in “one” solar system-the premise works if set in a binary star system with each sun being at least four or five AU apart, orbiting at least one other star over twenty AU distant. Communication may or may not be faster than light, but travel times would be on the order of weeks or months to reach some planets. Most terraformed worlds are moons orbiting gas giants. The Core “planets” probably all orbit the same gas giant, which if at least the size of Jupiter it would count as a system of its own. The space vehicles seen probably use inertial or gravimetric drives (they certainly have artificial gravity) for propulsion and some form of fusion thruster for navigation.