I think the Separatist movement’s official “premise” is that the Republic is corrupt and they are trying to break away from the Republic. As to why they attack planets? I guess they attack the ones that are “Republic” planets – because in Episode III there’s the full-scale war going on.
The viceroy is just a whiny little pawn. I don’t know what he wants to gain, other than more power for his own Trade Federation. Dooku, however, doesn’t really believe in the official Separatist premise, because remember, he is in cahoots with Palpatine (established at the end of Ep. II). He’s the other Sith – and I think Dooku just thinks that helping Palpatine turn the Republic upside down and form an empire will give him more power, too. Dooku just wants more Sith power, and he’s helping his master achieve it. Little does he know that Palpatine is setting him up to be whacked by a bigger better potential Sith apprentice (Anakin).
The separatists are a movement formed by Dooku under a false premise to help Palpatine achieve his evil Sith objective – turning the Republic into his own empire. The rebels are a movement that forms after the establishment of the empire, to attempt to fight back against it – probably to restore the republic form of government, if possible.