Does my character inherently know what the traits of each monster type are (such as vermin or undead being immune to mind-affecting spells) or do I need to make a knowledge roll for that?
Knowledge about creature types is not inherent
This has been clarified by James Jacobs (Creative Director):
So my thought is this: one roll. You simply declare "I want to figure out what this thing is. I rolled a 14." Then the DM asks what the relevant knowledge bonus is and determines if I know what I'm looking at. If it's, say, an aberration and I didn't bother with ranks in dungeoneering, I get nothing. But if it's an outsider and I have ranks in planes, then I might.
This is how I've always run it, and it works pretty well. Although usually I simply cut to the chase and ask the player to roll whatever the actually required Knowledge check is. This does mean that the player gets a clue as to what the monster's type is... if that's a problem (as in the case of an undead masquerading as a living creature, for example), I'll be more cagey about it, maybe asking them to make the roll and then looking at their sheet or asking them for ALL their knowledge ranks.
Any knowledge about monsters require Knowledge checks. Common monsters will require a low DC check, and anyone can make Knowledge checks if the DC is 10 or lower, without requiring ranks. That means anyone can identify traits of a creature of common creatures such as goblins, orcs, zombies and such. The GM defines what is considered common or not, though.
Since identifying a creature requires a check, with that check comes the knowledge of the creature's type. Maybe that skeleton-looking creature is not actually undead, but an outsider instead. How could you tell the difference?
Even the weakest of outsiders are uncommon enough so someone without ranks couldn't identify them, unless they live in some place where they regularly see outsiders (like the demiplanes) as those are at least a DC 10 + CR to identify: Lemures CR 1, Dretch CR 2, Cassisian CR 2, Lantern Archon CR 2, Lyrankien CR 2, etc
For undead, many of them are under CR 1, as such, it will depend on how common the GM decides that each specific undead is, maybe zombies and skeletons are common, but ghouls are rare enough to ask for DC 10 + CR instead (on a failed check, maybe they are identified as zombies). In areas plagued by undead (such as Ustalav), or with close proximity to them (like Nex), or even areas ruled by undead (see Geb), you could say that all undead are DC 5 + CR instead of only skeletons and zombies. As such, anyone could identify all undead traits easily, because those are very common.
As for what information you can get from a creature's type: It's up to your GM.
A successful check allows you to remember a bit of useful information about that monster.
This means that your GM is the only person who can say what piece of information you can get from a knowledge check. While vulnerabilities and immunities are pretty common, there may be others that are far more important to know, such as a medusa's or basilisk's petrifying gaze, the ability to drain energy or even the fact that certain creatures are immortal if not killed in a very special way.
This is not spelled out in the rules, and will come down to table conventions. Some GMs may treat anything in any Bestiary as confidential info during a game session, others might treat it as open knowledge.
At my own tables, I treat all general rules information (i.e. monster type details, universal monster rules mechanics etc) as public knowledge, and any monster-specific information as being off-limits without a check. Even knowing for sure what monster type a given creature is should require a basic monster knowledge check.
In a similar fashion, Spellcraft covers knowledge of what spells do and are capable of, but players generally need to know the mechanics for their spells, and so access to this information is normally allowed without restriction. But, if an NPC is using an unusual or rare spell, I might not allow players to look the information up without a check.