FTL drives do not exist in the Firefly universe, Joss has pretty much confirmed that. Even the Alliance starships spend months or years traveling the system at sublight.
Serenity shows Earth launching dozens of massive multi-generational sleeper ships at Sublight towards the system that later becomes their home, much of the 500 years between now and the show is the journey there, with over 100 years of terraforming and industrial development, meaning about 400 to get there. At 25% the speed of light thats 100 light years, at 50% c thats 200 light years, Hubble and the NGC catalouge currently list all *stars* within a 250 light year radius, but we know little or nothing about the planets (and number thereof) around them.
The Alliance occupies a system made up of dozens of planets, hundreds of moons, according to Serenity there is even a stable pattern to their orbits and at least 6 were immediately Earth type and orbited in the stars habitable zone, the others were heavily terraformed, the border worlds never achieving full vegetation becoming desert/old west arid type worlds.
Serenity herself, the ship, is powered by a "Photo-gravitic Pulse drive" that seems to involve a nuclear reactor, heating a gaseous material to a plasma state, spooling that material up and ejecting it to create an enormous amount of thrust that can propel her the distance equivalent to Earth/Mars separation or more in under three weeks.