If I multiclass a Magus and a Sorcerer in Pathfinder, how do my casting mechanics end up? Do I get a certain amount of slots that I must prepare in advance in addition to a few I can improvise with, or do they all become either prepared or improvised? Do my spell lists stay separate for the different slots, or can they cross over between caster lists?
You retain the casting mechanics of both classes separately.
If you were a multiclassed* 4th-level character, for example, a Magus 2/Sorcerer 2, you would have all the class abilities of the level you reached in each class, for instance a 2nd-level Magus and a 2nd-level Sorcerer. Their spell lists do not mix; you will have to prepare your Magus spells as if a 2nd-level Magus, and your known Sorcerer spells, numbering as many as a 2nd-level Sorcerer would know, can only be cast from the slots that a 2nd-level Sorcerer would have, and would suffer arcane spell failure chance appropriate for the armour you wear.
*(See Multiclassing header)