According to Pathfinder rules on conditions: "If more than one condition affects a character, apply them all. If effects can't combine, apply the most severe effect."
Using this, is it safe to assume that someone that if someone was intimidated (Applying the "Shaken" condition) then immediately succumbed to a Doom spell, (Also applying the "Shaken Condition) is this person now "Panicked" until the intimidate wears off? What if someone casts Doom on someone twice? Are they "Panicked" as well?
There's two different questions here - what conditions in general "don't stack" and you just take the worst, and which conditions "double down" when reapplied, and then which one of these ways does fear work? Fear doubles down; in other words it stacks and worsens down the condition track.
Fear
They don't list it in the condition summary, but under the Fear section you can see that someone who is shaken that gets shaken again moves up to frightened, etc. That's a special sort of stacking that only fear and a few other things do. Note that in several places, like the Intimidate skill, they deliberately nerf this by saying "two shakens caused by Intimidate doesn't stack to frightened" - they only say this because usually they do.
So yes, specifically, if you Intimidate someone and they're shaken, and then cast Doom on them and shake them again, they are frightened (not panicked, you'd need another for that - a second Doom could do it, but not another Intimidate because Intimidate explicitly doesn't stack).
Other Stuff
Normally conditions don't stack, but this does require interpretation at times. You don't Dazzle someone twice and then then become Blinded, but also they don't explicitly say Dazzled is a lesser condition than Blinded. In general it's easy when it's all numerical - "-2 morale penalty to attack rolls" and "-1 morale penalty to attack rolls" nets you just the -2. It's "obvious" that dazzled shouldn't stack with blinded even though the penalties aren't technically of the same type (-1 penalty to attack vs 50% miss chance) but that's something you/your GM just has to be smart about.
That's not how I read it. I think that if a character is both dazzled and blinded, only the blinded effect applies. Likewise fatigued and exhausted - only exhausted applies. (though fatigued specifically mentions that two fatigued effects turn into exhausted)
These effects can't combine because they're not compatible, so only the more severe effect applies.
I generally find it helpful to think of them as overlapping effects as opposed to stacking. Sometimes, as mentioned by @lizardggY, receiving the same condition multiple times will advance you along the condition path. In those cases it will be explicitly stated in the condition/power/feat/ability/spell/dance/jiggle.
So in your situation the shaken condition does not specify that multiple shaken conditions advance you to panicked. Nor does the panicked condition state that you can receive the condition from having multiple instances of shaken. Bit of belt and suspenders reasoning there, but still worth checking.
Duration and abilities that remove spell affects are where it becomes necessary to track each individually. So the description of Intimidate(Demoralize) states that the duration is 1 round, plus some more if you beat the DC well enough). Let's assume you do really stinking well and the duration is 3 rounds. The Doom spell has a duration of 1 minute per level, let's assume minimum caster level so 1 minute or 10 rounds.
In this situation you are under the affects of both Demoralize and Doom. Since they both produce the same affect you suffer from the standard Shaken penalties. Three rounds later the Demoralize wears off and you lose an instance of the Shaken condition. You still have 7 rounds of Doom, which means you are still Shaken.
Alternatively, let's assume in round 2 you are blessed with a Dispel Magic and your castor makes the check. This removes the Doom, however due to Demoralize you are still Shaken for one more round.
This is why it is always important to maintain your list of all affects, and sources, even if they produce the same condition.
