The Separatists (public) goals is to make a new pan-galactic state that is free of the problems the Republic had by 33 BBY* Corruption in the galactic Senate mostly. And they're not all bad guys. The Clone Wars cartoon has about 7 episodes that deal with the political side of things. The social and economic upheavals that the clone war either caused or made worse. That some of the seperatists were snarling supervillians but others were good people fed-up with the Republic bull-crap. However the snarling supervillians are in on the gag that their really there to give Palpatine a bogey-man to give him the leverage to total control of the Republic's government.
However all those politico's never had one VASTLY important advantage that Palpatine had. The Force. You know that thing that let's it's users see the future, read minds, control weaker minds, and throw lightning from your hands, and let a 65 yearold man wipe the floor with Samuel L. Jackson. Any question you could ask about how Palpatine could pull off making the entire galaxy his butt-monkey can be answered simply with, the Force. In otherwords, best McGuffin ever! Also the Sith were purposefully hiding themselfs from everyone.
Because Palpatine was a smart dictator. He sold the massacre of Jedi as a just action for their rebellion against the Republic.
*Before the Battle of Yavin. The accepted way to recon time in the Star Wars universe is the destruction of the first Death Star in Episode IV became the new year zero.
Bookmarks